Optical low-pass filter and optical apparatus having the same

ABSTRACT

An optical low-pass filter comprises a phase advancing area which advances a phase of a wavefront of an incident pencil of rays with respect to a phase of a wavefront of a center of the incident pencil of rays, and a phase retarding area which retards the phase of the wavefront of the incident pencil of rays with respect to the phase of the wavefront of the center of the incident pencil of rays, the phase advancing area and the phase retarding area alternately existing in the optical low-pass filter, and an optical apparatus comprises an image forming optical system, an image pickup element, and such an optical low-pass filter.

This is a Divisional Application of U.S. patent application Ser. No.08/804,155, now U.S. Pat. No. 6,144,493, filed Feb. 20, 1997.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to an optical low-pass filter suitable foruse with an optical apparatus having an image pickup element, such as avideo camera or a digital camera.

2. Description of Related Art

If an object having a periodical structure whose frequency is higherthan a frequency corresponding to the pixel period of an image pickupelement such as a CCD is to be photographed by using an opticalapparatus having such an image pickup element, for example, a videocamera or a digital camera, the image pickup element will output a falsesignal or a false color and the phenomenon of a degradation of aphotographed image occurs.

To prevent this phenomenon, it is known to employ an optical low-passfilter which separates an image of an object into two or more images andcuts the high-frequency component of the image by making use of thedouble refraction of a crystal plate.

If a sufficient low-pass effect is to be achieved by such opticallow-pass filter, the optical low-pass filter needs to have at least twocrystal plates. However, the use of expensive crystal plates leads tothe problem of an increase in cost. Furthermore, since the action ofcrystal does not have an appropriate effect on a polarization type ofobject, the problem of a decrease in the low-pass effect will occur.

To cope with these problems, Japanese Patent Publication No. Sho 44-1155discloses an optical low-pass filter which separates a wavefront intotwo or more images by means of a plurality of prisms.

However, the optical low-pass filter disclosed in Japanese PatentPublication No. Sho 44-1155 still has a problem. As shown in FIG. 1, theoptical low-pass filter can separate an image into two images on animage plane IS1, but there is another image plane IS2 on which the twoimages are mixed into one image. In an optical apparatus provided withan AF mechanism which determines the state of focus on the basis of thestate of an image plane, the AF mechanism determines the image plane IS2as an in-focus position, with the result that the low-pass effectdecreases.

BRIEF SUMMARY OF THE INVENTION

It is, therefore, an object of the present invention to provide anoptical low-pass filter which, when it is used with an opticalapparatus, can exhibit a stable low-pass effect and provide an goodimage.

To achieve the above object, in accordance with the present invention,there is provided an optical low-pass filter which comprises a phaseadvancing area which advances a phase of a wavefront of an incidentpencil of rays with respect to a phase of a wavefront of a center of theincident pencil of rays, and a phase retarding area which retards thephase of the wavefront of the incident pencil of rays with respect tothe phase of the wavefront of the center of the incident pencil of rays,the phase advancing area and the phase retarding area alternatelyexisting in the optical low-pass filter.

In accordance with another aspect of the present invention, there isprovided an optical low-pass filter which comprises a phase advancingarea which advances a phase of a wavefront of an incident pencil of rayswith respect to a phase of a wavefront of a center of the incidentpencil of rays, and a phase retarding area which retards the phase ofthe wavefront of the incident pencil of rays with respect to the phaseof the wavefront of the center of the incident pencil of rays, at leasteither one of the phase advancing area and the phase retarding areabeing provided as a plurality of phase advancing areas or a plurality ofphase retarding areas in the optical low-pass filter.

In accordance with another aspect of the present invention, there isprovided an optical low-pass filter which comprises an area having anoptical path length longer than an optical path length of a center ofthe optical low-pass filter, and an area having an optical path lengthshorter than the optical path length of the center of the opticallow-pass filter, the area having the longer optical path length and thearea having the shorter optical path length alternately existing in theoptical low-pass filter.

In accordance with another aspect of the present invention, there isprovided a method of manufacturing an optical low-pass filter, whichcomprises the steps of charging a material into a mold and removing thematerial molded by the mold, the optical low-pass filter alternatelyincluding a phase advancing area which advances a phase of a wavefrontof an incident pencil of rays with respect to a phase of a wavefront ofa center of the incident pencil of rays, and a phase retarding areawhich retards the phase of the wavefront of the incident pencil of rayswith respect to the phase of the wavefront of the center of the incidentpencil of rays, and the mold having a shape which corrects an erroroccurring during molding of the optical low-pass filter.

In accordance with another aspect of the present invention, there isprovided an optical apparatus which comprises an image forming opticalsystem, an image pickup element, and an optical low-pass filter, theoptical low-pass filter alternately including a phase advancing areawhich advances a phase of a wavefront of an incident pencil of rays withrespect to a phase of a wavefront of a center of the incident pencil ofrays, and a phase retarding area which retards the phase of thewavefront of the incident pencil of rays with respect to the phase ofthe wavefront of the center of the incident pencil of rays.

Various embodiments of the optical low-pass filter according to thepresent invention and the optical apparatus using such optical low-passfilter will be described later in detail with reference to theaccompanying drawings.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

FIG. 1 is a view aiding in describing the problem of an optical low-passfilter which makes use of a conventional prism;

FIG. 2 is a view aiding in describing the coordinate system of an imageplane;

FIG. 3 is a view showing a cylindrical coordinate system;

FIG. 4 is a view aiding in describing the distance between point imagesin an image plane and the distance from the center of the image plane toeach of the point images;

FIG. 5 is a view aiding in describing the distance between point imagesin an image plane and the distance from the center of the image plane toeach of the point images;

FIG. 6 is a diagrammatic view of the essential portion of aphotographing optical system according to Embodiment 1;

FIG. 7 is a view showing the contour lines of an optical low-pass filteraccording to a first numerical example;

FIG. 8 is a view showing the contour lines of the shape of an opticallow-pass filter which includes one phase advancing area and one phaseretarding area;

FIG. 9 is a view showing the contour lines of the shape of an opticallow-pass filter which includes only phase retarding areas;

FIG. 10 is a cross-sectional view of the shape of the optical low-passfilter according to the first numerical example;

FIG. 11 is a view showing the contour lines of a wavefront aberration inan exit pupil in the first numerical example;

FIG. 12 is a view showing a point spread in an image plane in the firstnumerical example;

FIG. 13 is a view showing a line spread of the first numerical exampleat F2.8;

FIG. 14 is a graph showing the MTF curve of the first numerical exampleat F2.8;

FIG. 15 is a view showing the contour lines of the shape of an opticallow-pass filter according to a second numerical example;

FIG. 16 is a cross-sectional view of the shape of the optical low-passfilter according to the second numerical example;

FIG. 17 is a view showing the contour lines of a wavefront aberration inan exit pupil in the second numerical example;

FIG. 18 is a view showing a point spread in an image plane in the secondnumerical example;

FIG. 19 is a view showing a line spread at F1.65 of the second numericalexample;

FIG. 20 is a graph showing the MTF curve of the second numerical exampleat F1.65;

FIG. 21 is a graph showing the MTF curve of the second numerical exampleat F5.6;

FIG. 22 is a view showing a wavefront aberration in an exit pupil whichan actual photographing optical system has;

FIG. 23 is a view showing the contour lines of a wavefront aberration inthe exit pupil of an optical system having the wavefront aberrationshown in FIG. 22 which optical system is provided with the opticallow-pass filter according to the second numerical example;

FIG. 24 is a view showing a point spread in the image plane of theoptical system having the wavefront aberration shown in FIG. 22 whichoptical system is provided with the optical low-pass filter according tothe second numerical example;

FIG. 25 is a graph showing the MTF curve at F1.65 of the optical systemhaving the wavefront aberration shown in FIG. 22 which optical system isprovided with the optical low-pass filter according to the secondnumerical example;

FIG. 26 is a graph showing the MTF curve at F5.6 of the optical systemhaving the wavefront aberration shown in FIG. 22 which optical system isprovided with the optical low-pass filter according to the secondnumerical example;

FIG. 27 is a view showing the contour lines of an optical low-passfilter having another shape;

FIG. 28 is a view showing the contour lines of an optical low-passfilter having another shape;

FIG. 29 is a view showing the contour lines of an optical low-passfilter having another shape;

FIG. 30 is a diagrammatic view showing the essential portion of aphotographing optical system according to Embodiment 2;

FIG. 31 is a diagrammatic view showing the essential portion of anotherphotographing optical system according to Embodiment 2;

FIG. 32 is a diagrammatic view showing the essential portion of aphotographing optical system according to Embodiment 3;

FIG. 33 is a diagrammatic view showing the essential portion of aphotographing optical system according to Embodiment 4;

FIG. 34 is a diagrammatic view showing the essential portion of aphotographing optical system according to Embodiment 5;

FIG. 35 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 5 as viewed from an object planeside;

FIG. 36 is a view showing the contour lines of the shape of the opticallow-pass filter according to Embodiment 5 as viewed from an image planeside;

FIG. 37 is a cross-sectional view taken in the direction indicated by anarrow “x” in FIG. 35;

FIG. 38 is a view showing the contour lines of a wavefront aberration inan exit pupil in Embodiment 5;

FIG. 39 is a diagrammatic view showing the essential portion of aphotographing optical system according to Embodiment 6;

FIG. 40 is a graph showing the MTF curve of Embodiment 6 for a cutofffrequency of 110 lines/mm;

FIG. 41 is a graph showing the MTF curve of Embodiment 6 for a cutofffrequency of 80 lines/mm;

FIG. 42 is a view showing the gradient refractive index of an opticallow-pass filter according to Embodiment 7;

FIG. 43 is a view showing the contour lines of a wavefront aberration inan exit pupil in Embodiment 7;

FIG. 44 is a view showing a point spread in an image plane in Embodiment7;

FIG. 45 is a view showing a line spread of Embodiment 7 at F2.8;

FIG. 46 is a graph showing the MTF curve of Embodiment 7 at F2.8;

FIG. 47 is a diagrammatic view showing the essential portion of aphotographing optical system according to Embodiment 8;

FIG. 48 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 8;

FIG. 49 is a view showing the variation in shape of the optical low-passfilter according to Embodiment 8, relative to the rotational directionof the optical low-pass filter;

FIG. 50 is a view showing a wavefront aberration obtained on a shorterfocal length side of the photographing optical system according toEmbodiment 8;

FIG. 51 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 8;

FIG. 52 is a view showing a relative line spread obtained on the shorterfocal length side of the photographing optical system according toEmbodiment 8;

FIG. 53 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment8;

FIG. 54 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 8;

FIG. 55 is a view showing a relative point spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 8;

FIG. 56 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 8;

FIG. 57 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment8;

FIG. 58 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 8, which is not provided with an optical low-pass filter;

FIG. 59 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 8, which is not provided with an optical low-pass filter;

FIG. 60 is a view showing a relative line spread obtained on the shorterfocal length side of the photographing optical system according toEmbodiment 8, which is not provided with an optical low-pass filter;

FIG. 61 is a graph showing an MTF curve obtained on a longer focallength side of the photographing optical system according to Embodiment8, which is not provided with an optical low-pass filter;

FIG. 62 is a view showing a wavefront aberration obtained on the longerfocal length side of the photographing optical system according toEmbodiment 8, which is not provided with an optical low-pass filter;

FIG. 63 is a view showing a relative point spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 8, which is not provided with an optical low-pass filter;

FIG. 64 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 8, which is not provided with an optical low-pass filter;

FIG. 65 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment8, which is not provided with an optical low-pass filter;

FIG. 66 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 9;

FIG. 67 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 9;

FIG. 68 is a view showing a relative line spread obtained on the shorterfocal length side of the photographing optical system according toEmbodiment 9;

FIG. 69 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment9;

FIG. 70 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 9;

FIG. 71 is a view showing a relative point spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 9;

FIG. 72 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 9;

FIG. 73 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment9;

FIG. 74 is a diagrammatic view showing the essential portion of aphotographing optical system having another arrangement;

FIG. 75 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according to a firstexample of Embodiment 10;

FIG. 76 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto the first example of Embodiment 10;

FIG. 77 is a view showing a relative line spread obtained on the shorterfocal length side of the photographing optical system according to thefirst example of Embodiment 10;

FIG. 78 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to the firstexample of Embodiment 10;

FIG. 79 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according to asecond example of Embodiment 10;

FIG. 80 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto the second example of Embodiment 10;

FIG. 81 is a view showing a relative line spread obtained on the shorterfocal length side of the photographing optical system according to thesecond example of Embodiment 10;

FIG. 82 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to the secondexample of Embodiment 10;

FIG. 83 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 11;

FIG. 84 is a view showing the variation in shape of the optical low-passfilter according to Embodiment 11, relative to the rotational directionof the optical low-pass filter;

FIG. 85 is a view showing a wavefront aberration obtained on a shorterfocal length side of the photographing optical system according toEmbodiment 11;

FIG. 86 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 11;

FIG. 87 is a view showing a relative line spread obtained on the shorterfocal length side of the photographing optical system according toEmbodiment 11;

FIG. 88 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment11;

FIG. 89 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 11;

FIG. 90 is a view showing a relative point spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 11;

FIG. 91 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 11;

FIG. 92 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment11;

FIG. 93 is a diagrammatic view showing the essential portion of aphotographing optical system according to Embodiment 12;

FIG. 94 is a view showing a wavefront aberration obtained on a shorterfocal length side of the photographing optical system according toEmbodiment 12;

FIG. 95 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 12;

FIG. 96 is a view showing a relative line spread obtained on the shorterfocal length side of the photographing optical system according toEmbodiment 12;

FIG. 97 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment12;

FIG. 98 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 13;

FIG. 99 is a view showing the variation in shape of the optical low-passfilter according to Embodiment 13, relative to the rotational directionof the optical low-pass filter;

FIG. 100 is a view showing a wavefront aberration obtained on a shorterfocal length side of the photographing optical system according toEmbodiment 13;

FIG. 101 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 13;

FIG. 102 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 13;

FIG. 103 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment13;

FIG. 104 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 13;

FIG. 105 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 13;

FIG. 106 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 13;

FIG. 107 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment13;

FIG. 108 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 14;

FIG. 109 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 14;

FIG. 110 is a view showing a relative line spread obtained on a shorterfocal length side of the photographing optical system according toEmbodiment 14;

FIG. 111 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment14;

FIG. 112 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 14;

FIG. 113 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 14;

FIG. 114 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 14;

FIG. 115 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment14;

FIG. 116 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 15;

FIG. 117 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 15;

FIG. 118 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 15;

FIG. 119 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment15;

FIG. 120 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 15;

FIG. 121 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 15;

FIG. 122 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 15;

FIG. 123 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment15;

FIG. 124 is a diagrammatic view showing the essential portion of aphotographing optical system according to Embodiment 16;

FIG. 125 is a view showing a wavefront aberration of the photographingoptical system according to Embodiment 16;

FIG. 126 is a view showing a relative point spread of the photographingoptical system according to Embodiment 16;

FIG. 127 is a view showing a relative line spread of the photographingoptical system according to Embodiment 16;

FIG. 128 is a graph showing an MTF curve of the photographing opticalsystem according to Embodiment 16;

FIG. 129 is a view showing a wavefront aberration of a photographingoptical system according to Embodiment 16, which is not provided with anoptical low-pass filter;

FIG. 130 is a view showing a relative point spread of the photographingoptical system according to Embodiment 16, which is not provided with anoptical low-pass filter;

FIG. 131 is a view showing a relative line spread of the photographingoptical system according to Embodiment 16, which is not provided with anoptical low-pass filter;

FIG. 132 is a graph showing an MTF curve of the photographing opticalsystem according to Embodiment 16, which is not provided with an opticallow-pass filter;

FIG. 133 is a view showing a wavefront aberration of a photographingoptical system according to a first example of Embodiment 17;

FIG. 134 is a view showing a relative point spread of the photographingoptical system according to the first example of Embodiment 17;

FIG. 135 is a view showing a relative line spread of the photographingoptical system according to the first example of Embodiment 17;

FIG. 136 is a graph showing an MTF curve of the photographing opticalsystem according to the first example of Embodiment 17;

FIG. 137 is a view showing a wavefront aberration of a photographingoptical system according to a second example of Embodiment 17;

FIG. 138 is a view showing a relative point spread of the photographingoptical system according to the second example of Embodiment 17;

FIG. 139 is a view showing a relative line spread of the photographingoptical system according to the second example of Embodiment 17;

FIG. 140 is a graph showing an MTF curve of the photographing opticalsystem according to the second example of Embodiment 17;

FIG. 141 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 18;

FIG. 142 is a view showing the variation in shape of the opticallow-pass filter according to Embodiment 18, relative to the rotationaldirection of the optical low-pass filter;

FIG. 143 is a view showing a wavefront aberration of the photographingoptical system according to Embodiment 18;

FIG. 144 is a view showing a relative point spread of the photographingoptical system according to Embodiment 18;

FIG. 145 is a view showing a relative line spread of the photographingoptical system according to Embodiment 18;

FIG. 146 is a graph showing an MTF curve of the photographing opticalsystem according to Embodiment 18;

FIG. 147 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 19;

FIG. 148 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 19;

FIG. 149 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 19;

FIG. 150 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment19;

FIG. 151 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 19;

FIG. 152 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 19;

FIG. 153 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 19;

FIG. 154 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment19;

FIG. 155 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 20;

FIG. 156 is a view showing the variation in shape of the opticallow-pass filter according to Embodiment 20, relative to the rotationaldirection of the optical low-pass filter;

FIG. 157 is a view showing a wavefront aberration obtained on a shorterfocal length side of the photographing optical system according toEmbodiment 20;

FIG. 158 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 20;

FIG. 159 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 20;

FIG. 160 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment20;

FIG. 161 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 20;

FIG. 162 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 20;

FIG. 163 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 20;

FIG. 164 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment20;

FIG. 165 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 21;

FIG. 166 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 21;

FIG. 167 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 21;

FIG. 168 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment21;

FIG. 169 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 21;

FIG. 170 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 21;

FIG. 171 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 21;

FIG. 172 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment21;

FIG. 173 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 22;

FIG. 174 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 22;

FIG. 175 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 22;

FIG. 176 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment22;

FIG. 177 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 22;

FIG. 178 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 22;

FIG. 179 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 22;

FIG. 180 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment22;

FIG. 181 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 23;

FIG. 182 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 23;

FIG. 183 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 23;

FIG. 184 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment23;

FIG. 185 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 23;

FIG. 186 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 23;

FIG. 187 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 23;

FIG. 188 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment23;

FIG. 189 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 24;

FIG. 190 is a view showing the variation in shape of the opticallow-pass filter according to Embodiment 24, relative to the rotationaldirection of the optical low-pass filter;

FIG. 191 is a view showing a wavefront aberration obtained on a shorterfocal length side of the photographing optical system according toEmbodiment 24;

FIG. 192 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 24;

FIG. 193 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 24;

FIG. 194 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment24;

FIG. 195 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 24;

FIG. 196 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 24;

FIG. 197 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 24;

FIG. 198 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment24;

FIG. 199 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 25;

FIG. 200 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 25;

FIG. 201 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 25;

FIG. 202 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment25;

FIG. 203 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 25;

FIG. 204 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 25;

FIG. 205 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 25;

FIG. 206 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment25;

FIG. 207 is a view showing the contour lines of the shape of an opticallow-pass filter according to a first example of Embodiment 26;

FIG. 208 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according to thefirst example of Embodiment 26;

FIG. 209 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto the first example of Embodiment 26;

FIG. 210 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto the first example of Embodiment 26;

FIG. 211 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to the firstexample of Embodiment 26;

FIG. 212 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according to thefirst example of Embodiment 26;

FIG. 213 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto the first example of Embodiment 26;

FIG. 214 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according to thefirst example of Embodiment 26;

FIG. 215 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to the firstexample of Embodiment 26;

FIG. 216 is a view showing the contour lines of the shape of an opticallow-pass filter according to a second example of Embodiment 26;

FIG. 217 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according to thesecond example of Embodiment 26;

FIG. 218 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto the second example of Embodiment 26;

FIG. 219 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto the second example of Embodiment 26;

FIG. 220 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to the secondexample of Embodiment 26;

FIG. 221 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 27;

FIG. 222 is a cross-sectional view showing the shape of the opticallow-pass filter according to Embodiment 27;

FIG. 223 is a view showing a wavefront aberration obtained at F1.65 on ashorter focal length side of a photographing optical system according toEmbodiment 27;

FIG. 224 is a view showing a relative point spread obtained at F1.65 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 27;

FIG. 225 is a view showing a relative line spread obtained at F1.65 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 27;

FIG. 226 is a graph showing an MTF curve obtained at F1.65 on theshorter focal length side of the photographing optical system accordingto Embodiment 27;

FIG. 227 is a view showing a wavefront aberration obtained at F2.8 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 27;

FIG. 228 is a view showing a relative point spread obtained at F2.8 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 27;

FIG. 229 is a view showing a relative line spread obtained at F2.8 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 27;

FIG. 230 is a graph showing an MTF curve obtained at F2.8 on the shorterfocal length side of the photographing optical system according toEmbodiment 27;

FIG. 231 is a view showing a wavefront aberration obtained at F5.6 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 27;

FIG. 232 is a view showing a relative point spread obtained at F5.6 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 27;

FIG. 233 is a view showing a relative line spread obtained at F5.6 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 27;

FIG. 234 is a graph showing an MTF curve obtained at F5.6 on the shorterfocal length side of the photographing optical system according toEmbodiment 27;

FIG. 235 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 28;

FIG. 236 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 28;

FIG. 237 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 28;

FIG. 238 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment28;

FIG. 239 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 28;

FIG. 240 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 28;

FIG. 241 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 28;

FIG. 242 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment28;

FIG. 243 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 29;

FIG. 244 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 29;

FIG. 245 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 29;

FIG. 246 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment29;

FIG. 247 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 29;

FIG. 248 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 29;

FIG. 249 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 29;

FIG. 250 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment29;

FIG. 251 is a view showing a wavefront aberration of a photographingoptical system according to Embodiment 30;

FIG. 252 is a view showing a relative point spread of the photographingoptical system according to Embodiment 30;

FIG. 253 is a view showing a relative line spread of the photographingoptical system according to Embodiment 30;

FIG. 254 is a graph showing an MTF curve of the photographing opticalsystem according to Embodiment 30;

FIG. 255 is a view showing a wavefront aberration of a photographingoptical system according to a first example of Embodiment 31;

FIG. 256 is a view showing a relative point spread of the photographingoptical system according to the first example of Embodiment 31;

FIG. 257 is a view showing a relative line spread of the photographingoptical system according to the first example of Embodiment 31;

FIG. 258 is a graph showing an MTF curve of the photographing opticalsystem according to the first example of Embodiment 31;

FIG. 259 is a view showing a wavefront aberration of a photographingoptical system according to a second example of Embodiment 31;

FIG. 260 is a view showing a relative point spread of the photographingoptical system according to the second example of Embodiment 31;

FIG. 261 is a view showing a relative line spread of the photographingoptical system according to the second example of Embodiment 31;

FIG. 262 is a graph showing an MTF curve of the photographing opticalsystem according to the second example of Embodiment 31;

FIG. 263 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 32;

FIG. 264 is a cross-sectional view showing the shape of the opticallow-pass filter according to Embodiment 32;

FIG. 265 is a view showing a wavefront aberration of a photographingoptical system according to Embodiment 32;

FIG. 266 is a view showing a relative point spread of the photographingoptical system according to Embodiment 32;

FIG. 267 is a view showing a relative line spread of the photographingoptical system according to Embodiment 32;

FIG. 268 is a graph showing an MTF curve of the photographing opticalsystem according to Embodiment 32;

FIG. 269 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 33;

FIG. 270 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 33;

FIG. 271 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 33;

FIG. 272 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment33;

FIG. 273 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 33;

FIG. 274 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 33;

FIG. 275 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 33;

FIG. 276 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment33;

FIG. 277 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 34;

FIG. 278 is a view showing the variation in shape of the opticallow-pass filter according to Embodiment 34, relative to the rotationaldirection of the optical low-pass filter;

FIG. 279 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 34;

FIG. 280 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 34;

FIG. 281 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 34;

FIG. 282 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment34;

FIG. 283 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 34;

FIG. 284 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 34;

FIG. 285 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 34;

FIG. 286 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment34;

FIG. 287 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 35;

FIG. 288 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 35;

FIG. 289 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 35;

FIG. 290 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment35;

FIG. 291 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 35;

FIG. 292 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 35;

FIG. 293 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 35;

FIG. 294 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment35;

FIG. 295 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 36;

FIG. 296 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 36;

FIG. 297 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 36;

FIG. 298 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment36;

FIG. 299 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 36;

FIG. 300 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 36;

FIG. 301 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 36;

FIG. 302 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment36;

FIG. 303 is a view showing a wavefront aberration of a photographingoptical system according to Embodiment 37;

FIG. 304 is a view showing a relative point spread of the photographingoptical system according to Embodiment 37;

FIG. 305 is a view showing a relative line spread of the photographingoptical system according to Embodiment 37;

FIG. 306 is a graph showing an MTF curve of the photographing opticalsystem according to Embodiment 37;

FIG. 307 is a view showing a wavefront aberration of a photographingoptical system according to a first example of Embodiment 38;

FIG. 308 is a view showing a relative point spread of the photographingoptical system according to the first example of Embodiment 38;

FIG. 309 is a view showing a relative line spread of the photographingoptical system according to the first example of Embodiment 38;

FIG. 310 is a graph showing an MTF curve of the photographing opticalsystem according to the first example of Embodiment 38;

FIG. 311 is a view showing a wavefront aberration of a photographingoptical system according to a second example of Embodiment 38;

FIG. 312 is a view showing a relative point spread of the photographingoptical system according to the second example of Embodiment 38;

FIG. 313 is a view showing a relative line spread of the photographingoptical system according to the second example of Embodiment 38;

FIG. 314 is a graph showing an MTF curve of the photographing opticalsystem according to the second example of Embodiment 38;

FIG. 315 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 39;

FIG. 316 is a cross-sectional view showing the shape of the opticallow-pass filter according to Embodiment 39;

FIG. 317 is a view showing a wavefront aberration of a photographingoptical system according to Embodiment 39;

FIG. 318 is a view showing a relative point spread of the photographingoptical system according to Embodiment 39;

FIG. 319 is a view showing a relative line spread of the photographingoptical system according to Embodiment 39;

FIG. 320 is a graph showing an MTF curve of the photographing opticalsystem according to Embodiment 39;

FIG. 321 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 40;

FIG. 322 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 40;

FIG. 323 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 40;

FIG. 324 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment40;

FIG. 325 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 40;

FIG. 326 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 40;

FIG. 327 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 40;

FIG. 328 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment40;

FIG. 329 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 41;

FIG. 330 is a view showing a wavefront aberration obtained at F1.65 on ashorter focal length side of a photographing optical system according toEmbodiment 41;

FIG. 331 is a view showing a relative point spread obtained at F1.65 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 41;

FIG. 332 is a view showing a relative line spread obtained at F1.65 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 41;

FIG. 333 is a graph showing an MTF curve obtained at F1.65 on theshorter focal length side of the photographing optical system accordingto Embodiment 41;

FIG. 334 is a view showing a wavefront aberration obtained at F2.8 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 41;

FIG. 335 is a view showing a relative point spread obtained at F2.8 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 41;

FIG. 336 is a view showing a relative line spread obtained at F2.8 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 41;

FIG. 337 is a graph showing an MTF curve obtained at F2.8 on the shorterfocal length side of the photographing optical system according toEmbodiment 41;

FIG. 338 is a view showing a wavefront aberration obtained at F5.6 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 41;

FIG. 339 is a view showing a relative point spread obtained at F5.6 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 41;

FIG. 340 is a view showing a relative line spread obtained at F5.6 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 41;

FIG. 341 is a graph showing an MTF curve obtained at F5.6 on the shorterfocal length side of the photographing optical system according toEmbodiment 41;

FIG. 342 is a view showing a wavefront aberration obtained at F1.65 on alonger focal length side of the photographing optical system accordingto Embodiment 41;

FIG. 343 is a view showing a relative point spread obtained at F1.65 onthe longer focal length side of the photographing optical systemaccording to Embodiment 41;

FIG. 344 is a view showing a relative line spread obtained at F1.65 onthe longer focal length side of the photographing optical systemaccording to Embodiment 41;

FIG. 345 is a graph showing an MTF curve obtained at F1.65 on the longerfocal length side of the photographing optical system according toEmbodiment 41;

FIG. 346 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 42;

FIG. 347 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 42;

FIG. 348 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 42;

FIG. 349 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment42;

FIG. 350 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 42;

FIG. 351 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 42;

FIG. 352 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 42;

FIG. 353 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment42;

FIG. 354 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 43;

FIG. 355 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 43;

FIG. 356 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 43;

FIG. 357 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment43;

FIG. 358 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 43;

FIG. 359 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 43;

FIG. 360 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 43;

FIG. 361 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment43;

FIG. 362 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 44;

FIG. 363 is a view showing a wavefront aberration obtained at F1.65 on ashorter focal length side of a photographing optical system according toEmbodiment 44;

FIG. 364 is a view showing a relative point spread obtained at F1.65 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 44;

FIG. 365 is a view showing a relative line spread obtained at F1.65 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 44;

FIG. 366 is a graph showing an MTF curve obtained at F1.65 on theshorter focal length side of the photographing optical system accordingto Embodiment 44;

FIG. 367 is a view showing a wavefront aberration obtained at F2.8 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 44;

FIG. 368 is a view showing a relative point spread obtained at F2.8 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 44;

FIG. 369 is a view showing a relative line spread obtained at F2.8 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 44;

FIG. 370 is a graph showing an MTF curve obtained at F2.8 on the shorterfocal length side of the photographing optical system according toEmbodiment 44;

FIG. 371 is a view showing a wavefront aberration obtained at F5.6 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 44;

FIG. 372 is a view showing a relative point spread obtained at F5.6 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 44;

FIG. 373 is a view showing a relative line spread obtained at F5.6 onthe shorter focal length side of the photographing optical systemaccording to Embodiment 44;

FIG. 374 is a graph showing an MTF curve obtained at F5.6 on the shorterfocal length side of the photographing optical system according toEmbodiment 44;

FIG. 375 is a view showing a wavefront aberration obtained at F1.65 on alonger focal length side of the photographing optical system accordingto Embodiment 44;

FIG. 376 is a view showing a relative point spread obtained at F1.65 onthe longer focal length side of the photographing optical systemaccording to Embodiment 44;

FIG. 377 is a view showing a relative line spread obtained at F1.65 onthe longer focal length side of the photographing optical systemaccording to Embodiment 44;

FIG. 378 is a graph showing an MTF curve obtained at F1.65 on the longerfocal length side of the photographing optical system according toEmbodiment 44;

FIG. 379 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 45;

FIG. 380 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 45;

FIG. 381 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 45;

FIG. 382 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment45;

FIG. 383 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 45;

FIG. 384 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 45;

FIG. 385 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 45;

FIG. 386 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment45;

FIG. 387 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 46;

FIG. 388 is a cross-sectional view of the basic shape of the opticallow-pass filter according to Embodiment 46;

FIG. 389 is a cross-sectional view of the actual shape of the opticallow-pass filter according to Embodiment 46.

FIG. 390 is a view showing a relative point spread in the image plane ofa photographing optical system according to Embodiment 46;

FIG. 391 is a view showing the relative line spread at F1.65 of thephotographing optical system according to Embodiment 46;

FIG. 392 is a view showing the MTF curve at F1.65 of the photographingoptical system according to Embodiment 46;

FIG. 393 is a graph showing the MTF curve at F5.6 of the photographingoptical system according to Embodiment 46;

FIG. 394 is a diagrammatic view showing the essential portion of aphotographing optical system according to Embodiment 47;

FIG. 395 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 47;

FIG. 396 is a view showing a wavefront aberration of the photographingoptical system according to Embodiment 47;

FIG. 397 is a view showing a relative point spread of the photographingoptical system according to Embodiment 47;

FIG. 398 is a view showing a relative line spread of the photographingoptical system according to Embodiment 47;

FIG. 399 is a graph showing an MTF curve of the photographing opticalsystem according to Embodiment 47;

FIG. 400 is a view showing a wavefront aberration of a photographingoptical system according to Embodiment 47, which is not provided with anoptical low-pass filter;

FIG. 401 is a view showing a relative point spread of the photographingoptical system according to Embodiment 47, which is not provided with anoptical low-pass filter;

FIG. 402 is a view showing a relative line spread of the photographingoptical system according to Embodiment 47, which is not provided with anoptical low-pass filter;

FIG. 403 is a graph showing an MTF curve of the photographing opticalsystem according to Embodiment 47, which is not provided with an opticallow-pass filter;

FIG. 404 is a view showing in contour line an example of deformation ofa surface r1 which occurs during molding;

FIG. 405 is a view showing in contour line an example of deformation ofa surface r2 which occurs during molding;

FIG. 406 is a view showing in contour line an example of non-uniformdistribution of an inner refractive index which occurs during molding;

FIG. 407 is a view showing the contour lines of a transmitted wavefrontobtainable when the amounts of errors shown in FIGS. 405 and 406 areadded to the surface r2 in Embodiment 47;

FIG. 408 is a view showing in contour line the deviation of the surfacer1 from the spheric shape thereof, which deviation is intended tocorrect the amounts of errors shown in FIGS. 405 and 406 in Embodiment47;

FIG. 409 is a view showing in contour line the deviation of the surfacer1 from the spheric shape thereof, a shape for correcting the amounts oferrors shown in FIGS. 405 and 406 and a low-pass shape being added tothe surface r1 according to Embodiment 47;

FIG. 410 is a view showing the contour lines of the shape of an opticallow-pass filter according to Embodiment 48;

FIG. 411 is a view showing a wavefront aberration obtained on a shorterfocal length side of a photographing optical system according toEmbodiment 48;

FIG. 412 is a view showing a relative point spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 48;

FIG. 413 is a view showing a relative line spread obtained on theshorter focal length side of the photographing optical system accordingto Embodiment 48;

FIG. 414 is a graph showing an MTF curve obtained on the shorter focallength side of the photographing optical system according to Embodiment48;

FIG. 415 is a view showing a wavefront aberration obtained on a longerfocal length side of the photographing optical system according toEmbodiment 48;

FIG. 416 is a view showing a relative point spread obtained on thelonger focal length side of the photographing optical system accordingto Embodiment 48;

FIG. 417 is a view showing a relative line spread obtained on the longerfocal length side of the photographing optical system according toEmbodiment 48;

FIG. 418 is a graph showing an MTF curve obtained on the longer focallength side of the photographing optical system according to Embodiment48;

FIG. 419 is a view showing in contour line the deviation of a surfacer14 from the spheric shape thereof, a shape for correcting the amount oferror and a low-pass shape being added to the surface r14 according toEmbodiment 48; and

FIG. 420 is a diagrammatic view showing the essential portion of anoptical apparatus which includes a photographing optical systemaccording to any of the embodiments of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

(Embodiment 1)

FIG. 2 is a view aiding in describing the coordinate system of an imageplane.

In the present embodiment, as shown in FIG. 2, an amplitude distributionobtainable when a point image is separated into four point images in animage plane is expressed as the following cylindrical coordinate system:

U(r, θ)=U(r)×cos (2θ+δ),  (1)

where δ is a constant.

The shape of an optical low-pass filter which is represented by acylindrical coordinate system is obtained from the shape of a wavefront(aberration) which gives that amplitude distribution.

The procedure for obtaining the shape of such optical low-pass filterwill be described below. First of all, the shape of an optical low-passfilter which is represented by the cylindrical coordinate system shownin FIG. 3 is expressed as:

S(ρ,φ)=A×R(ρ)×cos (mφ+δ),  (2)

where A and δ are constants, ρ(0≦ρ≦1) is a coordinate system relative toa radial direction of the optical low-pass filter, which coordinatesystem is standardized on a radius “a” of the optical low-pass filter,R(ρ) is the radial shape of the optical low-pass filter, φ(0≦φ≦2π) is acoordinate system relative to the rotational direction of the opticallow-pass filter, and m is defined as m=2, 3, 4, . . . (an integer) onthe basis of the periodicity of the optical low-pass filter relative tothe rotational direction. In the present embodiment, δ=0.

Letting n be the refractive index of the optical low-pass filter, awavefront aberration to be given to a transmitted wavefront by theaforesaid shape is expressed as: $\begin{matrix}\begin{matrix}{{{W\left( {\rho,\varphi} \right)} = {{S\left( {\rho,\varphi} \right)} \times \left( {1 - n} \right)}},} \\{{= {A^{\prime} \times {R(\rho)} \times \cos \quad \left( {m\quad \varphi} \right)}},}\end{matrix} & (3)\end{matrix}$

where A′=(1−n)A.

It is known that an amplitude distribution in an image plane due to awavefront having such wavefront aberration is obtained by thediffraction integral of a pupil function. As can be seen from“Principles of optics” written by M. Born and E. Wolf, Section 9.4, pp473-478 (5th Edition 1975), the diffraction integral becomes:$\begin{matrix}{{{U\left( {u,v,\theta} \right)} = {C{\int_{0}^{1}{\int_{0}^{2\pi}{^{{\{{{{kA}^{\prime}{R{(\rho)}}{\cos {({m\quad \varphi})}}} - {v\quad {{\rho cos}{({\varphi - \theta})}}} - {\frac{1}{2}u\quad \rho^{2}}}\}}}\quad \rho {\rho}\quad {\varphi}}}}}},} & (4)\end{matrix}$

where C is a constant, k is a wave number (k=2π/λ), λ is a wavelength,and (u, v, θ) are standardized cylindrical coordinates in the imageplane, u representing a coordinate axis relative to the direction of anoptical axis, v representing a coordinate axis relative to a radialdirection, and θ representing a coordinate axis relative to a rotationaldirection.

If an object is at infinity, the following relation is establishedbetween v and actual coordinates:

v=πr/λF,  (5)

where r is an actual distance and F is the F number of a photographingoptical system.

If the distribution of the amplitude in a reference image plane isconsidered, u=0 and Expression (4) is re-written as: $\begin{matrix}{{U\left( {v,\theta} \right)} = {C{\int_{0}^{1}{\int_{0}^{2\pi}{^{{\{{{{kA}^{\prime}{R{(\rho)}}{\cos {({m\quad \varphi})}}} - {v\quad {{\rho cos}{({\varphi - \theta})}}}}\}}}\quad \rho {\rho}\quad {{\varphi}.}}}}}} & (6)\end{matrix}$

In accordance with §9.4 of “Principles of Optics”, Expression (6) isre-written as follows by using the Jacobi identity: $\begin{matrix}{{{U\left( {v,\theta} \right)} = {4C{\underset{s = 0}{\overset{\infty}{\sum^{\prime}}}\quad {\underset{s^{\prime} = 0}{\overset{\infty}{\sum^{\prime}}}\quad {\left( {- i} \right)^{s}\left( {- i} \right)^{s^{\prime}}{\int_{0}^{1}{\int_{0}^{2\pi}{J_{s}\left\{ {{kA}^{\prime}{R(\rho)}} \right\} {J_{s^{\prime}}\left( {v\quad \rho} \right)}\cos \quad \left( {m\quad s\quad \varphi} \right)\cos \quad \left( {s^{\prime}\left( {\varphi - \theta} \right)} \right\} \rho \quad {\rho}\quad {\varphi}}}}}}}}},} & (7)\end{matrix}$

where Js(x) is a Bessel function of the first kind.

Further, it follows from a termwise integration for φ that:$\begin{matrix}\begin{matrix}{{U\left( {v,\theta} \right)} = \quad {4C{\underset{s = 0}{\overset{\infty}{\sum^{\prime}}}\quad {\left( {- i} \right)^{{({m - 1})}s}\cos \quad \left( {m\quad s\quad \theta} \right){\int_{0}^{1}{J_{s}\left\{ {{kA}^{\prime}{R(\rho)}} \right\} {J_{m\quad s}\left( {v\quad \rho} \right)}\rho \quad {\rho}}}}}}} \\{= \quad {4{C\left\lbrack {{\frac{1}{2}{\int_{0}^{1}{J_{o}\left\{ {{kA}^{\prime}{R(\rho)}} \right\} {J_{o}\left( {v\quad \rho} \right)}\rho \quad {\rho}}}} +} \right.}}} \\{\quad {{\left( {- i} \right)^{({m - 1})}\cos \quad \left( {m\quad \theta} \right){\int_{0}^{1}{J_{1}\left\{ {{kA}^{\prime}{R(\rho)}} \right\} {J_{m}\left( {v\quad \rho} \right)}\rho \quad {\rho}}}} +}} \\{\quad {{\left( {- i} \right)^{2{({m - 1})}}\cos \quad \left( {2m\quad \theta} \right){\int_{0}^{1}{J_{2}\left\{ {{kA}^{\prime}{R(\rho)}} \right\} {J_{2m}\left( {v\quad \rho} \right)}\rho \quad {\rho}}}} +}} \\{\left. \quad {{\left( {- i} \right)^{3{({m - 1})}}\cos \quad \left( {3m\quad \theta} \right){\int_{0}^{1}{J_{3}\left\{ {{kA}^{\prime}{R(\rho)}} \right\} {J_{3m}\left( {v\quad \rho} \right)}\rho \quad {\rho}}}} + \ldots}\quad \right\rbrack.}\end{matrix} & (8)\end{matrix}$

The first term of Expression (8) represents an amplitude which isdistributed uniformly in the rotational direction with respect to theorigin of the cylindrical coordinates, and the second term and thefollowing represent an amplitude which is distributed periodically inthe rotational direction. From the integral of the second term, it canbe seen that an amplitude (or intensity) distribution occurs with aperiod of cos (mθ) in the rotational direction of the image plane.

Specifically, it can be seen that if an optical low-pass filter having aplurality of areas formed continuously in the rotational direction ofits opening portion is provided in a photographing optical system, someof the areas serving to exert a phase advancing action on the wavefrontof an incident pencil of rays so as to advance the phase of thewavefront of the incident pencil of rays with respect to the phase ofthe wavefront at the center of the opening portion, and the other areasserving to exert a phase retarding action on the wavefront of theincident pencil of rays so as to retard the phase of the wavefront ofthe incident pencil of rays with respect to the phase of-the wavefrontat the center of the opening portion, the wavefront of the pencil ofrays transmitted through the optical system can be given a phasevariation for phase advance or retardation, so that its Fraunhoferdiffraction image can be formed as a plurality of separate spots in animage forming plane in which the pencil of rays is focused.

The optical low-pass filter of the present invention, like conventionaloptical low-pass filters, has the effect of making MTF zero for apredetermined spatial frequency and decreasing MTF relative to spatialfrequencies higher than the predetermined spatial frequency, but can beinexpensively manufactured without the need to use an expensivematerial. In addition, it is desirable to dispose the optical low-passfilter of the present invention in the neighborhood of the stop of aphotographing optical system since an equal phase advancing/retardingaction can be exerted on pencils of rays each having a different imageheight.

In Expression (8), letting the second term represent the amplitudedistribution shown by Expression (1), m of cos (mθ) becomes 2.Therefore, the shape of the optical low-pass filter is:

S(ρ,φ)=A×R(ρ)×cos (2φ).  (9)

The amplitude distribution expressed by Expression (8) is:$\begin{matrix}\begin{matrix}{{U\left( {v,\theta} \right)} = \quad {4C\left\lbrack {{\frac{1}{2}{\int_{0}^{1}{J_{o}\left\{ {{kA}^{\prime}{R(\rho)}} \right\} {J_{o}\left( {v\quad \rho} \right)}\rho \quad {\rho}}}} -} \right.}} \\{\quad {{i\quad \cos \quad 2\quad \theta {\int_{0}^{1}{J_{1}\left\{ {{kA}^{\prime}{R(\rho)}} \right\} {J_{2}\left( {v\quad \rho} \right)}\rho \quad {\rho}}}} -}} \\{\quad {{\cos \quad 4\quad \theta {\int_{0}^{1}{J_{2}\left\{ {{kA}^{\prime}{R(\rho)}} \right\} {J_{4}\left( {v\quad \rho} \right)}\rho \quad {\rho}}}} +}} \\{\quad {{i\quad \cos \quad 6\quad \theta {\int_{0}^{1}{J_{3}\left\{ {{kA}^{\prime}{R(\rho)}} \right\} {J_{6}\left( {v\quad \rho} \right)}\rho \quad {\rho}}}} + {\ldots \quad {\rbrack.}}}}\end{matrix} & (10)\end{matrix}$

It can be considered that if the amount of aberration (W) is small,almost all integral values are contained in the first several terms ofExpression (10) and the integral of the second term substantiallyrepresents the spread of the amplitude (=U×U*) which is producedperiodically in the rotational direction in the image plane. Therefore,the radial shape R(ρ) of the optical low-pass filter may be determinedso that the absolute value of the integral of the second term ofExpression (10) reaches its maximum value in the neighborhood of apredetermined position vc determined by the pixels of an image pickupelement, for example, a CCD: $\begin{matrix}{{{\int_{0}^{1}{J_{1}\left\{ {{kA}^{\prime}{R(\rho)}} \right\} {J_{2}\left( {v\quad \rho} \right)}\rho \quad {\rho}}}}_{v = {vc}}.} & (11)\end{matrix}$

If the pitch of the pixels of the CCD is p, the cutoff frequency of aluminance signal, which is required for the CCD, is fc=1/(2p), and adistance dc by which point images are separated from each other in thepixel-array direction of the CCD by the optical low-pass filter becomes:

dc=1/(2=fc)=p.  (12)

If the direction of separation of the point images intersects thepixel-array direction of the CCD as shown in FIG. 4, the followingrelation is obtained:

rc=dc/2.  (13)

If the direction of separation of the point images intersects thepixel-array direction of the CCD as shown in FIG. 5, the followingrelation is obtained:

rc=dc·({square root over (2)}/2).  (14)

Thus, vc in Expression (11) is obtained.

The integral of Expression (11) can reach its maximum when v=vc on thecondition that:

J ₁ {kA′R(ρ)}≈J ₂(vcρ).  (15)

If Expression (15) is satisfied, the optical low-pass filter accordingto the present invention can make MTF approximately zero at apredetermined spatial frequency even in a case where an aperturediameter varies according to photographing conditions.

In the low-pass filter according to the present invention, if adirection in which its phase advancing or retarding action is large isinclined in the range of 30° to 60°, most preferably by an angle of 45°,with respect to the direction of array of pixels of the CCD, a linespread relative to a direction perpendicular to the direction of arrayof pixels of the CCD is averaged, so that the low-pass effect of thelow-pass filter can be effectively improved.

The low-pass filter according to the present invention can be formed tohave a complex curved shape, by molding a synthetic resin material(plastic material) such as acrylic resin or a glass material, or byforming such a synthetic resin material on a glass substrate.

One surface of a plane-parallel plate to be provided in the vicinity ofthe stop may be formed to have such a shape, or the shape may beseparated into two cylindrical shapes into which to form two surfaces,respectively.

If the shape of the optical low-pass filter according to the presentinvention is added to a surface of a lens which constitutes part of thephotographing optical system, it is possible to realize an opticalelement having both the function of a lens and the function of anoptical low-pass filter.

The optical low-pass filter according to the present invention may beformed in part of a variable-transmittance stop which is formed by an ECliquid crystal or the like and whose transmittance varies.

In addition, the shape of the optical low-pass filter according to thepresent invention may be formed in part of an infrared cut-filter whichconstitutes part of the photographing optical system, or the opticallow-pass filter itself may be formed of an infrared cutting material.

A sufficient low-pass effect can be achieved by using only the opticallow-pass filter according to the present invention. However, if theoptical low-pass filter according to the present invention is usedtogether with a crystal optical low-pass filter, it is possible to copewith different spatial frequencies to be cut off, by adjusting thethickness of crystal of the crystal optical low-pass filter.Accordingly, the optical low-pass filter according to the presentinvention can be applied to various optical apparatuses using differentimage pickup elements having different specifications.

By employing the optical low-pass filter according to the presentinvention, it is possible to set the MTF of an image forming opticalsystem to not less than 5% at a desired spatial frequency and to notless than 20% in a frequency range higher than the desired spatialfrequency.

FIG. 6 is a diagrammatic view of the essential portion of aphotographing optical system for an optical apparatus which employs anoptical low-pass filter according to Embodiment 1.

The shown photographing optical system includes lenses L1 to L9 whichconstitute an image forming optical system. The lenses L1 to L3constitute a first lens unit of positive refractive power, the lenses L4to L6 constitute a second lens unit of negative refractive power, thelens L7 constitutes a third lens unit of positive refractive power, andthe lenses L8 to L9 constitute a fourth lens group of positiverefractive power. In the photographing optical system according toEmbodiment 1, its magnification is varied by the second lens unit movingalong the optical axis of the photographing optical system, andcompensation for its image plane and focusing are effected by the fourthlens group moving along the optical axis.

The shown photographing optical system also includes an optical low-passfilter 1 according to Embodiment 1 of the present invention, a stop 2,and an infrared cut-filter 3. Reference numeral 4 denotes a CCD (imagepickup element). The optical low-pass filter 1 is provided in thevicinity of the stop 2 so that a phase change is equally given to thewavefronts of various pencils of rays having different angles of view soas to provide an effective low-pass effect.

The acting surface of the optical low-pass filter 1 which exerts a phaseadvancing/retarding action on the phase of the wavefront of an incidentpencil of rays is formed as a flat surface made of a synthetic resinmaterial such as acrylic resin. The shape of the acting surface is shownin FIG. 7 in contour line. As shown in FIG. 7, with respect to thecenter of the opening portion, each portion marked “+” is projected to amaximum degree, whereas each portion marked “−” is dented to a maximumdegree.

As shown in FIG. 7, if the shape of the optical low-pass filteraccording to Embodiment 1 of the present invention is represented in acylindrical coordinate system the origin of which corresponds to thecenter of the opening portion, the shape of the optical low-pass filtercontinuously changes in the rotational direction from a portion having aphase advancing action on the phase of an incident wavefront (either ofthe portions marked “−”) to a portion having a phase retarding action onthe phase of the incident wavefront (the adjacent one of the portionsmarked “+”).

The acting surface of the optical low-pass filter needs to have two ormore portions each having a phase advancing action and two or moreportions each having a phase retarding action. For example, in the caseof an acting surface which is only provided with one portion having aphase advancing action and one portion having a phase retarding actionas shown in FIG. 8, the shape of the acting surface contains a largeamount of slope component, so that the image in an image plane is onlydeviated in a particular direction without being effectively separatedinto point images. In the case of an acting surface which is composed ofonly portions having phase advancing actions or only portions havingphase retarding actions as shown in FIG. 9, an image can be separatedinto a plurality of images in a particular image plane. However, sincethere is necessarily an image plane in which separated images are fusedinto one image, the low-pass effect of such acting surface is loweredsimilarly to the optical low-pass filter disclosed in Japanese PatentPublication No. Sho 44-1155.

A first numerical example of the surface shape of the optical low-passfilter according to Embodiment 1 is shown below:

S 1(ρ, φ)=A 1 ×R 1(ρ)×cos (2φ),   (16)

where

R 1(ρ)=(2.622ρ−1.140ρ²)λ, 0≦ρ≦1, 0≦φ≦2π.  (17)

The cross-sectional shape of the acting surface of the first numericalexample for φ=0 is shown in FIG. 10.

The first numerical example of the optical low-pass filter according toEmbodiment 1 produces a wavefront aberration, such as that shown in FIG.11, which is analogous to the shape of the optical low-pass filter, inthe exit pupil of an ideal optical system composed of aberration-freelenses, so that point images are annularly spaced apart from one anotherin an image plane, as shown in FIG. 12. It is, therefore, possible todecrease MTF relative to high-frequency components to a further extent.FIG. 13 shows the line spread function (LSF) obtained by performing anaddition in a direction perpendicular to the pixel-array direction of aCCD, which is obtained at F2.8 from the first numerical example of theoptical low-pass filter, and FIG. 14 shows the MTF curve of the firstnumerical example of the optical low-pass filter.

FIG. 15 shows contour lines which represent the shape of the actingsurface of a second numerical example of the optical low-pass filteraccording to Embodiment 1. This shape is expressed as the followingexpression:

S 2(ρ, φ)=A 2 ×R 2(ρ)×cos (2φ),  (18)

where

R 2(ρ)=(3.534ρ+2.867ρ²−13.267ρ³−7.079ρ⁴+12.737ρ⁵)λ, 0≦ρ≦1, 0≦φ≦2π.  (19)

In the second numerical example, portions which has phase advancingactions on the phase of an incident wavefront and portions which hasphase retarding actions on the phase of the incident wavefront areprovided in the radial direction as well. The cross-sectional shape ofthe acting surface of the second numerical example for φ=0 is shown inFIG. 16. The radial shape of the second numerical example must be suchthat the region of high-frequency components to be cut off does not varywith a variation in the F number of the photographing optical system.For this reason, it is desirable to use a nonlinear radial shape such asthat shown in FIG. 15.

FIG. 17 shows the contour lines of a wavefront aberration in the exitpupil of the second numerical example, and FIG. 18 shows the contourlines of the point spread in its image plane. FIG. 19 shows the linespread obtained by performing an addition in a direction perpendicularto the pixel-array direction of a CCD, which is obtained in the imageplane at F1.65. FIGS. 20 and 21 are MTF curves for F1.65 and F5.6,respectively.

The above description has been given on the assumption that thephotographing optical system is composed of aberration-free lenses, buteven if aberration is present in a lens of the photographing opticalsystem, the effect of the optical low-pass filter according to thepresent invention is not impaired. From the characteristics of thephotographing optical system, letting Φ the wavefront aberration of alens system, a pupil function h0 of the lens system and a pupil functionh1 of the optical low-pass filter become:

h 0=exp(iΦ),  (20)

h 1=exp(iw(ρ,φ)),  (21)

and a pupil function h of the entire system becomes

h=h 0×h 1.  (22)

Since an amplitude distribution U is obtained as a Fourier transform Fof h,

U=F(h)=F(h 0×h 1)=F(h 0)*F(h 1),  (23)

where the operator * represents a convolution. Since F(h1) coincideswith the amplitude distribution due to each of the optical low-passfilters described previously, it is apparent that an equal low-passeffect is attained.

The optical low-pass filter of the second numerical example expressed byExpression (18) may be added to a photographing optical system havingthe wavefront aberration shown in FIG. 22. FIG. 23 shows the contourlines of a wavefront aberration in the exit pupil of this example, andFIG. 24 shows the contour lines of the point spread in its image plane.FIGS. 25 and 26 are MTF curves for F1.65 and F5.6, respectively.Incidentally, in accordance with such calculation results which takeaccount of the wavefront aberration of the photographing optical system,the photographing optical system is defocused along its optical axis sothat the low-frequency component of its MTF reaches a maximum.

FIGS. 27 and 28 show the contour lines of different optical low-passfilters having different shapes in their rotational directions.

The respective shapes shown in FIGS. 27 and 28 are expressed as:

S 3(ρ, φ)=A 3×R 3(ρ)×cos (3φ),  (24)

S 4(ρ, φ)=A 4×R 4(ρ)×cos (4φ).  (25)

From the integral of the second term of Expression (8), the opticallow-pass filter expressed by Expression (24) has the spreadcharacteristics of separating an image into six images in its imageplane, while the optical low-pass filter expressed by Expression (25)has the spread characteristics of separating an image into eight imagesin its image plane.

Any of the above-described examples of the optical low-pass filter has ashape which is periodical in the rotational direction (φ) and in whichportions having phase advancing actions and portions having phaseretarding actions are symmetrical in shape. However, in the case of anoptical low-pass filter having a shape which is asymmetrical andnon-periodical as shown by contour lines in FIG. 29 or a shape in whichportions which exert phase advancing/retarding actions are not uniformin height (depth), it is considered that a function which represents theshape of such a surface is not simple unlike any of the aforesaidfunctions and can be expressed like

S(ρ,φ)=ΣAm×Rm(ρ)×cos (mφ), m=1, 2, 3, . . . n,  (26)

and the result of Expression (8) which gives the diffraction integral ofExpression (26) shows that a plurality of spots formed in an image planeare impaired in periodicity relative to the rotational direction and inthe uniformity of their shapes.

Even in this case, if the shape of a line spread obtainable byintegrating intensity spread in a direction perpendicular to thepixel-array direction of a CCD is approximately equal to a rectangularspread having a width equivalent to a pitch p of the pixels of the CCD,the function of the optical low-pass filter is not affected. To realizesuch an optical low-pass filter, the integral of the second term ofExpression (8) needs only to have a limit value in the neighborhood ofvc (Expression (13) or (14)) determined by the pixel pitch p, withrespect to the wavefront aberration due to the shape represented byExpression (26).

(Embodiment 2)

FIG. 30 is a diagrammatic view of the essential portion of aphotographing optical system according to Embodiment 2.

In Embodiment 2, a surface having a low-pass effect is provided at thelocation of a surface of a lens which constitutes part of thephotographing optical system. In FIG. 30, identical reference numeralsare used to denote constituent elements whose functions are basicallyidentical to those of the corresponding ones of Embodiment 1, and thedescription thereof is omitted herein for the sake of simplicity.

It is desirable that a surface having a low-pass effect be provided on alens disposed in the vicinity of the stop 2. In Embodiment 2, such asurface is provided at the location of a surface r1 of a lens L7′. Theshape of this lens surface can be expressed as follows:

S(ρ,φ)=Sp(ρ)+Asp(ρ)+Lp(ρ,φ),  (27)

where Sp(ρ) represents the shape of a curved surface determined by aradius of the osculating sphere, “R”, Asp(ρ) represents theaxisymmetrical aspheric shape expressed by a polynomial of ρ or thelike, and Lp(ρ, φ) represents a surface shape having the low-pass effectaccording to the present invention. As can be seen from Expression (27),the term of Sp(ρ)+Asp(ρ) is calculated as ordinary lens design, and theterm of Lp(ρ, φ) is added to the term of Sp(ρ)+Asp(ρ) to determine thefinal shape of the lens surface.

Such a lens can readily be manufactured as a molded lens by melting andmolding a synthetic resin material such as acrylic resin or glass, as byusing a mold having a molding shape equivalent to the shape determinedby Expression (27). The lens may also be manufactured by polishing asurface of a glass lens into the spheric surface expressed by Sp(ρ) ofExpression (27) and covering the spheric surface with a synthetic resinmaterial, such as acrylic resin, to add to the spheric surface theaspheric surface shape and the surface shape having the low-pass effectboth of which are determined by Asp(ρ)+Lp(ρ, φ) of Expression (27).

It is possible to adopt another arrangement in which, as shown in FIG.31, a surface having a low-pass effect is provided at the location of asurface r2 of a lens L7″, whereas an axisymmetrical aspheric surface isprovided at the location of the surface r1 of the lens L7″.

(Embodiment 3)

FIG. 32 is a diagrammatic view of the essential portion of aphotographing optical system according to Embodiment 3.

In Embodiment 3, a surface having a low-pass effect according to thepresent invention is provided on a variable density element 5 whichactively varies its transmittance to vary the amount of light, and thevariable density element 5 is disposed in the vicinity of the stop 2.The variable density element 5 has a structure in which an EC liquidcrystal or the like is hermetically sealed between flat plates, and thesurface having the low-pass effect is formed on either of the flatplates. In FIG. 32, identical reference numerals are used to denoteconstituent elements whose functions are basically identical to those ofthe corresponding ones of Embodiment 1, and the description thereof isomitted herein for the sake of simplicity.

(Embodiment 4)

FIG. 33 is a diagrammatic view of the essential portion of aphotographing optical system according to Embodiment 4.

In Embodiment 4, a surface having a low-pass effect is provided on aninfrared cut-filter 30, and the infrared cut-filter 30 is disposed inthe vicinity of the stop 2 of the photographing optical system. Thesurface having the low-pass effect may be prepared by forming asynthetic resin material having an infrared cutting function, such asthat set forth in Japanese Laid-Open Patent Application No. Hei6-118228, into a shape such as that stated in Embodiment 1, or by addinga shape, such as that stated in Embodiment 1, to a flat infraredcut-filter by using a material different from the material of the flatinfrared cut-filter. In FIG. 33, identical reference numerals are usedto denote constituent elements whose functions are basically identicalto those of the corresponding ones of Embodiment 1, and the descriptionthereof is omitted herein for the sake of simplicity.

(Embodiment 5)

FIG. 34 is a diagrammatic view of the essential portion of aphotographing optical system according to Embodiment 5.

In Embodiment 5, surfaces which exert phase advancing/retarding actionson the phase of a wavefront are formed on the opposite sides of a flatplate made of a synthetic resin material such as acrylic resin, the flatplate being provided in the vicinity of the stop 2. The respectivesurfaces have cylindrical shapes as shown in FIGS. 35 and 36 in contourline, and the directions in which the respective surfaces haverefractive powers are rotated 90° with respect to each other. FIG. 37shows a cross-sectional shape taken in the direction indicated by anarrow “x” in FIG. 35.

Since the contour lines of the wavefront transmitted through an opticallow-pass filter 10 is transformed as shown in FIG. 38, the opticallow-pass filter 10 of Embodiment 5 achieves an effect similar to theabove-described optical low-pass filter 1 of Embodiment 1. In FIG. 34,identical reference numerals are used to denote constituent elementswhose functions are basically identical to those of the correspondingones of Embodiment 1, and the description thereof is omitted herein forthe sake of simplicity.

(Embodiment 6)

FIG. 39 is a diagrammatic view of the essential portion of aphotographing optical system according to Embodiment 6.

In Embodiment 6, the acting surface of the optical low-pass filter 1 isprovided in the vicinity of the stop 2, and a crystal plate 6 havinginclined double refraction axes is provided in the photographing opticalsystem. In FIG. 39, identical reference numerals are used to denoteconstituent elements whose functions are basically identical to those ofthe corresponding ones of Embodiment 1, and the description thereof isomitted herein for the sake of simplicity.

The optical low-pass filter 1 principally cuts a high frequencycomponent, and the crystal plate 6 sets the cutoff frequency requiredfor a CCD. This arrangement enables the photographing optical system tobe used with different CCDs each having a different number of CCDs.

FIGS. 40 and 41 show the respective MTF curves for optical cutofffrequencies of 110 lines/mm and 80 lines/mm in the above-describedarrangement. In each of FIGS. 40 and 41, a dotted curve represents theMTF curve of Embodiment 6 which uses the optical low-pass filter 1alone, while a solid curve represents the MTF curve of Embodiment 6which uses the crystal plate 6 in addition to the optical low-passfilter 1. The range denoted by an arrow A represents a domain in which ahigh-frequency suppression effect can be obtained, and symbol fc denotesthe optical cutoff frequency determined by the crystal plate 6.

(Embodiment 7)

Each of the embodiments 1 to 6 is arranged to provide a low-pass effectby giving a wavefront aberration to an incident pencil of rays while theincident pencil of rays is passing through areas of differentthicknesses of an optical low-pass filter. The wavefront aberration iscaused by variations in the optical path length of the passing pencil ofrays due to the different thicknesses of the respective areas of theoptical low-pass filter. The optical path length is given by the productof the distance of the path traversed in a medium by a pencil of rays(thickness d) and the refractive index of the medium. Accordingly, notonly by making the thickness d different for each of the areas of theoptical low-pass filter but also by making the refractive indexdifferent for each of the areas, it is possible to vary the optical pathlength and realize the optical low-pass filter according to the presentinvention.

Embodiment 7 relates to an optical low-pass filter having differentrefractive indices for different areas.

The gradient refractive index of the optical low-pass filter ofEmbodiment 7 which is represented by a cylindrical coordinate system isobtained from the shape of a wavefront (aberration) which gives a pointimage spread such as that shown in FIG. 2.

The procedure for obtaining such gradient refractive index will bedescribed below. The gradient refractive index of the optical low-passfilter which is represented by the cylindrical coordinate system isexpressed as:

N(ρ, φ)=N 0+Nr(ρ)×cos (mφ+δ),  (28)

where N0 is the refractive index of the central portion of the opticallow-pass filter, ρ is coordinates relative to the radial direction ofthe optical low-pass filter, Nr(ρ) is a gradient refractive indexrelative to the radial direction of the optical low-pass filter, φ iscoordinates relative to the rotational direction of the optical low-passfilter, m is defined as m=2, 3, 4, . . . (an integer) on the basis ofthe periodicity of the optical low-pass filter relative to therotational direction, and δ is a constant (in Embodiment 7, δ=0).

Letting d be the thickness of the optical low-pass filter, from suchgradient refractive index, the wavefront aberration given to atransmitted wavefront by the aforesaid shape is expressed as:$\begin{matrix}\begin{matrix}{{W\left( {\rho,\varphi} \right)} = {A \times \delta \quad {N\left( {\rho,\varphi} \right)} \times d}} \\{= {A \times {{Nr}(\rho)} \times \cos \quad \left( {m\quad \varphi} \right) \times d}} \\{= {A^{\prime} \times {{Nr}(\rho)} \times \cos \quad {\left( {m\quad \varphi} \right).}}}\end{matrix} & (29)\end{matrix}$

Assuming that the second term of Expression (8) represents the amplitudedistribution of Expression (1) referred to previously, it is desirableto set “m” of cos (mθ) to 2. Thus, the distribution of the refractiveindex error of the optical low-pass filter becomes:

δN(ρ, φ)=Nr(ρ)×cos (2φ).  (30)

A numerical example of an optical low-pass filter according toEmbodiment 7, which has a predetermined gradient refractive index, isshown below:

N 1(ρ, φ)=N 0+Nr 1(ρ)×cos (2φ),  (31)

where

N 1(ρ)=(2.184ρ−0.949ρ²)λ, 0≦ρ≦1, 0≦φ≦2π.  (32)

The gradient refractive index of the numerical example for φ=0 is shownin FIG. 42.

The numerical example of the optical low-pass filter according toEmbodiment 7 produces a wavefront aberration, such as that shown in FIG.43, which is analogous to the shape of the optical low-pass filter, inthe exit pupil of an ideal optical system composed of aberration-freelenses, so that point images are annularly separated from one another inan image plane, as shown in FIG. 44. It is, therefore, possible todecrease MTF relative to high-frequency components to a further extent.FIG. 45 shows the line spread obtained by performing an addition in adirection perpendicular to the pixel-array direction of a CCD, which isobtained at F2.8 from the numerical example of the optical low-passfilter, and FIG. 46 shows the MTF curve of the numerical example of theoptical low-pass filter.

In this manner, it is possible to obtain a desired low-pass effect bygiving a predetermined gradient refractive index to the optical low-passfilter.

As is apparent from the above description, an optical low-pass filterwhich makes the optical path length of its passing pencil of raysdifferent by means of its varied thickness (shape) so that some areas ofthe optical low-pass filter exert phase advancing actions and the otherareas exert phase retarding actions is equivalent in function and effectto an optical low-pass filter which makes the optical path length of itspassing pencil of rays different by means of its gradient refractiveindex so that some areas of the optical low-pass filter exert phaseadvancing actions and the other areas exert phase retarding actions.Specifically, any of the optical low-pass filters of Embodiments 1 to 6can be replaced with an optical low-pass filter having a gradientrefractive index.

It is also possible to adopt an optical low-pass filter which differ inboth thickness and refractive index for each area.

(Embodiment 8)

Embodiment 8 of the present invention is shown in FIG. 47. In Embodiment8, the shape of an optical low-pass filter which will be described lateris added to the shape of a surface of a lens which constitutes part ofthe photographing optical system shown in FIG. 47. The photographingoptical system includes the stop 2 and the infrared cut-filter 3.Reference numeral 4 denotes a CCD.

Lens data for the photographing optical system are shown in Table 1.

TABLE 1 Radius of Refractive Abbe Surface Curvature Separation IndexNumber First  1 47.30490 1.25000 1.847 23.9 Lens  2 24.99391 4.900001.603 31.1 Unit  3 422.44811 0.20000 1  4 22.00462 2.77000 1.697 55.1  552.79733 D1 Second  6 36.64051 0.60000 1.773 49.6 Lens  7 5.727662.89700 1 Unit  8 −11.03929 1.50000 1.530 55.5  9* 9.01249 1.06000 1 1012.47778 1.50000 1.847 23.9 11 82.77610 D2 1 Stop 12 0.00000 1.20000 1Third  13* 14.31644 3.40000 1.530 55.5 Lens 14 −30.92051 D3 1 UnitFourth 15 11.08409 0.80000 1.847 23.9 Lens 16 5.18699 5.20000 1.530 55.5Unit  17* −18.46439 1 f 4.12 66.48 mm F number 1.65 2.84 D1 0.950 23.416D2 23.716 1.250 D3 7.938 8.939 2ω 57.2° 3.9°

If the direction of the optical axis of the photographing optical systemand a direction perpendicular to the optical axis are respectively takenas a z-axis and an h-axis and the direction of propagation of light istaken as a positive direction, each axisymmetrical aspheric surface ofEmbodiment 8 is expressed by the following expression: $\begin{matrix}{{Z = {\frac{h^{2}/R}{1 + \sqrt{1 - {\left( {1 + k} \right)\left( {h/R} \right)^{2}}}} + {ah}^{4} + {bh}^{6} + {ch}^{8} + {dh}^{10}}},} & (33)\end{matrix}$

where R is a radius of the osculating surface and k, a, b, c and d areaspheric coefficients.

Data for the respective aspheric surfaces are shown in Table 2.

TABLE 2 Surface k a b c d 9 −4.91288e − 01 −7.61380e − 05 −6024942e − 06 8.03346e − 07 −2.42975e − 08 13 −9.08186e − 01 −6.54077e − 05  6.23762e− 08 −2.94373e − 09 −1.30582e − 11 17  6.23416e + 00 −2.18108e − 05 1.14897e − 06 −5.82846e − 07  1.72059e − 08

The photographing optical system (zoom lens) of Embodiment 8 includes afirst lens unit having a positive refractive power, a second lens unithaving a negative refractive power, a stop, a third lens unit having apositive refractive power, and a fourth lens unit having a positiverefractive power. The photographing optical system is arranged to varyits magnification by moving the second lens unit, and to effectcompensation for its image plane and focusing by moving the fourth lensunit.

The optical low-pass filter of Embodiment 8 is realized by adding anasymmetrical variation in shape to the radius of curvature (shown inTable 1) of a surface r14 of the third lens unit disposed in thevicinity of the stop 2.

If the coordinate system of the lens surface is a cylindrical coordinatesystem (ρ, φ) which is, as shown in FIG. 3, represented by thecoordinate system ρ relative to the radial direction and the coordinatesystem φ relative to the rotational direction, in each of which itsorigin corresponds to the center of the opening portion, a shape S(ρ, φ)of the optical low-pass filter of Embodiment 8 is expressed by thefollowing expressions: $\begin{matrix}{{{S\left( {\rho,\varphi} \right)} = {\sum\limits_{m}\quad {{Am} \times {{Rm}(\rho)} \times \cos \quad \left\{ {m\left( {\varphi + {\delta \quad m}} \right)} \right\}}}},{m = 2},6,10} & (34)\end{matrix}$

 R 2(ρ)=(a2ρ+b2ρ² +c2ρ³)λ,  (35)

R 6(ρ)=(a6ρ+b6ρ²)λ,  (36)

R 10(ρ)=(a10ρ+b10ρ²)λ.  (37)

As expressed by Expressions (34) to (37), the shape of the opticallow-pass filter of Embodiment 8 which is added to the third lens unit isasymmetrical about the axis thereof.

The coefficients used in Embodiment 8 are shown in Table 3.

TABLE 3 A₂ = A₆ = A₁₀ = 1 δ₂ = δ₆ = δ₁₀ = π/4 a₂ = 8.1687  a₆ = −0.0924a₁₀ = −0.0185 b₂ = −13.925 b₆ = −0.2358 b₁₀ = −0.0472 c₂ = 7.5154 

The contour lines of the optical low-pass filter of Embodiment 8(Expression (34)) are shown in FIG. 48, and the variation in shape ofthe optical low-pass filter relative to the rotational direction (the φdirection) for ρ=1 is shown in FIG. 49.

In the case of the optical low-pass filter which is realized as theamount of variation in shape of the lens, its wavefront aberration is:

W(ρ, φ)=S(ρ, φ)×(1−n),  (38)

where n is the refractive index of the lens.

The wavefront aberration W(ρ, φ) is expressed as: $\begin{matrix}{{{W\left( {\rho,\varphi} \right)} = {\sum\limits_{m}\quad {{Am} \times {{Wrm}(\rho)} \times \cos \quad \left\{ {m\left( {\varphi + {\delta \quad m}} \right)} \right\}}}},} & (39)\end{matrix}$

where m is an integer and Am and δm are constants.

FIGS. 50, 51, 52 and 53 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 54, 55, 56 and 57 respectively show a wavefrontaberration (λ=587.56 nm), a relative point spread (white), a relativeline spread (white) and an MTF curve (white) which are obtained on alonger focal length side of such photographing optical system. FIGS. 58,59, 60 and 61 respectively show a wavefront aberration (λ=587.56 nm), arelative point spread (white), a relative line spread (white) and an MTFcurve (white) which are obtained on a shorter focal length side of aphotographing optical system including no optical low-pass filter (Table1). FIGS. 62, 63, 64 and 65 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a longer focallength side of such photographing optical system.

As is apparent from the above description, the optical low-pass filterof Embodiment 8 produces a wavefront aberration analogous to its shapeto separate a point image into a plurality of point images in an imageplane so that the value of MTF can be effectively reduced over the rangeof spatial frequencies higher than a predetermined spatial frequency atwhich the value of MTF is made zero. The predetermined spatial frequency(cutoff frequency) at which the value of MTF is made zero is obtainedfrom the pitch of the pixels of an image pickup element such as a CCD tobe used. In Embodiment 8, the pixel pitch is 5 μm, and the cutofffrequency is 100 lines/mm.

(Embodiment 9)

In Embodiment 9, a shape for providing a low-pass effect is formed at asurface different from the surface r14.

Embodiment 9 uses the same functional expressions that express the shapeS(ρ, φ) of the optical low-pass filter of Embodiment 8, and thecoefficients used in Embodiment 9 are the same as those shown in Table3, except for the coefficient Am (in Embodiment 9, A₁₀=A₆=A₂=0.92). Theshape S(ρ, φ) of the optical low-pass filter is added to an asphericsurface r13 located in the vicinity of the stop 2. Lens data are thesame as those used in Embodiment 8. The shape of the optical low-passfilter of Embodiment 9 is such that the shape of the optical low-passfilter of Embodiment 8 is squeezed by a small amount in the direction ofthe z-axis (optical axis).

FIGS. 66, 67, 68 and 69 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 70, 71, 72 and 73 respectively show a wavefrontaberration (λ=587.56 nm), a relative point spread (white), a relativeline spread (white) and an MTF curve (white) which are obtained on alonger focal length side of such photographing optical system.

As is apparent from the above description, the optical low-pass filtermay be added to any surface that is located in the vicinity of the stop2. As shown in FIG. 74, a flat plate 5 which does not greatly affect thephotographing optical system and to which the optical low-pass filter isadded may be provided in the vicinity of the stop 2.

(Embodiment 10)

In Embodiment 10, the setting of the angle δ is changed.

Embodiment 10 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 8, and thecoefficients used in Embodiment 10 are the same as those shown in Table3, except for the coefficient A (in Embodiment 10, A=0.73). Lens dataare the same as those used in Embodiment 8.

FIGS. 75, 76, 77 and 78 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of a photographing optical system including an opticallow-pass filter of δ=30°.

FIGS. 79, 80, 81 and 82 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of a photographing optical system including an opticallow-pass filter of δ=22.5° and A=0.9.

(Embodiment 11)

The shape S(ρ, φ) of the optical low-pass filter of Embodiment 11 isexpressed by the following expressions: $\begin{matrix}{{{S\left( {\rho,\varphi} \right)} = {\sum\limits_{m}\quad {{Am} \times {{Rm}(\rho)} \times \cos \quad \left\{ {m\left( {\varphi + {\delta \quad m}} \right)} \right\}}}},{m = 2},6,10,14,} & (40)\end{matrix}$

 R 2(ρ)=(a2ρ+b2ρ² +c2ρ³)λ,  (41)

R 6(ρ)=(a6ρ+b6ρ²)λ,  (42)

R 10(ρ)=(a10ρ+n10ρ²)λ,  (43)

R 14(ρ)=(a14ρ+b14ρ²)λ.  (44)

The coefficients used in Embodiment 11 are shown in Table 4.

TABLE 4 A₂ = A₆ = A₁₀ = δ₂ = δ₆ = δ₁₀ = A₁₄ = 1 δ₁₄ = π/4 a₂ =  4.7591a₆ = −0.069311 a₁₀ = −0.023104 a₁₄ = −0.0033005 b₂ = −8.1207 b₆ =−0.17685 b₁₀ = −0.058950 b₁₄ = −0.0084211 c₂ =  4.3586

The contour lines of the optical low-pass filter of Embodiment 11(Expression (40)) are shown in FIG. 83, and the variation in shape ofthe optical low-pass filter relative to the rotational direction (the φdirection) for ρ=1 is shown in FIG. 84.

FIGS. 85, 86, 87 and 88 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of a photographing optical system in which the opticallow-pass filter having the aforesaid shape is added to the surface r14represented by the corresponding lens data shown in Embodiment 8. FIGS.89, 90, 91 and 92 respectively show a wavefront aberration (λ=587.56nm), a relative point spread (white), a relative line spread (white) andan MTF curve (white) which are obtained on a longer focal length side ofsuch photographing optical system. As can be seen from FIGS. 85 to 92,Embodiment 11 can achieve an effect similar to the optical low-passfilter of Embodiment 8.

(Embodiment 12)

FIG. 93 shows Embodiment 12 of the present invention. In Embodiment 12,an optical low-pass filter having a gradient refractive index accordingto the present invention is added to a photographing optical systemwhich includes the optical low-pass filter 1, the stop 2 and theinfrared cut-filter 3. In FIG. 93, reference numeral 4 denotes a CCD.

Lens data for Embodiment 12 are shown in Table 5.

TABLE 5 Radius of Refractive Abbe Surface Curvature Separation IndexNumber First  1 47.30490 1.25000 1.847 23.9 Lens  2 24.99391 4.900001.603 31.1 Unit  3 −422.44811 0.20000 1  4 22.00462 277000 1.697 55.1  552.79733 D1 1 Second  6 36.64051 0.60000 1.773 49.6 Lens  7 5.727662.89700 1 Unit  8 −11.03929 1.50000 1.530 55.5  9* 9.01249 1.06000 1 1012.47778 1.50000 1.847 23.9 11 82.77610 D2 1 LPF 12 0.00000 1.000001.492 57.4 13 0.00000 0.00000 1 Stop 14 0.00000 1.20000 1 Third  15*14.31644 3.40000 1.530 55.5 Lens 16 −30.92051 D3 1 Unit Fourth 1711.08409 0.80000 1.847 23.9 Lens 18 5.18699 5.20000 1.530 55.5 Unit  19*−18.46439 4.00726 1 f 4.12 66.48 F number 1.65 2.843 D1 0.950 23.416 D223.046 0.580 D3 7.938 8.939 2ω 57.2° 3.9°

If the direction of the optical axis of the photographing optical systemand a direction perpendicular to the optical axis are respectively takenas a z-axis and an h-axis and the direction of propagation of light istaken as a positive direction, each axisymmetrical aspheric surface ofEmbodiment 12 is expressed by the following expression: $\begin{matrix}{{Z = {\frac{h^{2}/R}{1 + \sqrt{1 - {\left( {1 + k} \right)\left( {h/R} \right)^{2}}}} + {ah}^{4} + {bh}^{4} + {ch}^{8} + {dh}^{10}}},} & (45)\end{matrix}$

where R is the radius of the osculating surface and k, a, b, c and areaspheric coefficients.

Data for the respective aspheric surfaces are shown in Table 6.

TABLE 6 Surface k a b c d 9 −4.91288e − 01 −7.61380e − 05 −6.24942e − 06 8.03346e − 07 −2.43975e − 08 15 −9.08186e − 01 −6.54077e − 05  6.23762e− 08 −2.94373e − 09 −1.30582e − 11 19  6.23416e + 00 −2.18108e − 05 1.14897e − 06 −5.82846e − 07  1.72059e − 08

The optical low-pass filter of Embodiment 12 is realized by adding aflat filter at the position of the stop 2 which is represented by thecorresponding lens data shown in Embodiment 8.

If the coordinate system of the opening portion of the optical low-passfilter is the cylindrical coordinate system (ρ, φ) shown in FIG. 3, agradient refractive index N(ρ, φ) of the optical low-pass filter ofEmbodiment 12 is expressed by the following expressions: $\begin{matrix}{\begin{matrix}{{{N\left( {\rho,\varphi} \right)} = {{N0} + {\delta \quad {N\left( {\rho,\varphi} \right)}}}},} \\{{= {{N0} + {\sum\quad {{Am} \times {{Nrm}(\rho)} \times \cos \quad \left\{ {m\left( {\varphi + {\delta \quad m}} \right)} \right\}}}}},}\end{matrix}{{m = 2},6,10,}} & (46)\end{matrix}$

 Nr 2(ρ)=(a2ρ+b2ρ² +c2ρ³)λ,  (47)

Nr 6(ρ)=(a6ρ+b6ρ²)λ,  (48)

Nr 10(ρ)=(a10ρ+b10ρ²)λ.  (49)

The coefficients used in Embodiment 12 are shown in Table 7.

TABLE 7 A₂ = A₆ = A₁₀ = 1 δ₂ = δ₆ = δ₁₀ = π/4 a₂ = 9.5893  a₆ = −0.0850a₁₀ = −0.0170 b₂ = −16.300 b₆ = −0.2169 b₁₀ = −0.0434 c₂ =8.9421 

In the case of the optical low-pass filter which is realized by giving avariation in refractive index to an optical member, its wavefrontaberration is:

W(ρ, φ)=δN(ρ, φ)×d,  (50)

where d is the thickness of the optical low-pass filter.

FIGS. 94, 95, 96 and 97 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter.

As is apparent from the above description of Embodiment 12, with anoptical member having a gradient refractive index, it is possible togive a wavefront aberration to an incident pencil of rays while theincident pencil of rays is passing through different positions of theoptical member. Accordingly, the optical member can be made to functionas the optical low-pass filter according to the present invention.

The use of such a gradient refractive index is not limited to Embodiment12, and the gradient refractive index can be used to produce a wavefrontaberration which would be produced by the optical low-pass filter of anyother embodiment.

In other words, the optical low-pass filter of the present invention canbe realized only if one optical low-pass filter includes an area havinga short optical path length (an area which advances the phase of anincident pencil of rays with respect to the central phase thereof) andan area having a long optical path length with respect to the opticalpath length of the center of the incident pencil of rays passing throughthe optical low-pass filter. Since the optical path length is given bythe product of a distance “1” of the path traversed in a medium by apencil of rays and a refractive index “n” of the medium, it is possibleto set the optical path length to a desired value by making either orboth of the distance “1” and the refractive index “n” different fromthose at the center.

(Embodiment 13)

Embodiment 13 of the present invention will be described below.

The optical low-pass filter of Embodiment 13 is added to the surface r14of the third lens unit shown in FIG. 47 similarly to the opticallow-pass filter of Embodiment 8, but Embodiment 13 differs in shape fromEmbodiment 8.

The shape S(ρ, φ) of the optical low-pass filter of Embodiment 13 isexpressed by the following expressions:

S(ρ, φ)=R(ρ)×cos {2(φ+kρ+δ)},  (51)

R(ρ)=(aρ+bρ ² +cρ ³)λ.  (52)

The coefficients used in Embodiment 13 are shown in Table 8.

TABLE 8 A = 1. a = 4.0558  b = −6.4442 c = 3.7941 δ = 1/4π κ = −1/8π

The contour lines of the optical low-pass filter of Embodiment 13(Expression (51)) are shown in FIG. 98, and the variation in shape ofthe optical low-pass filter relative to the rotational direction (the φdirection) for ρ=0.5, 1 is shown in FIG. 99.

In the case of the optical low-pass filter which is realized as theamount of variation in shape of the lens, its wavefront aberration is:

W(ρ, φ)=S(ρ, φ)×(1−n),  (53)

where n is the refractive index of the lens.

The wavefront aberration W(ρ, φ) which occurs in the optical low-passfilter of Embodiment 13 is expressed as:

W(ρ, φ)=Wr(ρ)×cos [m{φ+kf(ρ)+δ}],  (54)

where m is an integer not less than 2, k and δ are constants, and f(ρ)is an arbitrary function of ρ.

FIGS. 100, 101, 102 and 103 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread. (white), a relative line spread(white) and an MTF curve (white) relative to the x-direction, all ofwhich are obtained on a shorter focal length side of the photographingoptical system including the optical low-pass filter. FIGS. 104, 105,106 and 107 respectively show a wavefront aberration (λ=587.56 nm), arelative point spread (white), a relative line spread (white) and an MTFcurve (white) which are obtained on a longer focal length side of suchphotographing optical system.

As is apparent from the above description, the optical low-pass filterof Embodiment 13 produces a wavefront aberration analogous to its shapeto separate a point image into a plurality of point images in an imageplane so that the value of MTF can be effectively reduced over the rangeof spatial frequencies higher than a predetermined spatial frequency atwhich the value of MTF is made zero. The predetermined spatial frequency(cutoff frequency) at which the value of MTF is made zero is obtainedfrom the pitch of the pixels of an image pickup element such as a CCD tobe used. In Embodiment 13, the pixel pitch is 5 μm, and the cutofffrequency is 100 lines/mm.

(Embodiment 14)

In Embodiment 14 of the present invention, the cutoff frequency is setto a lower frequency than the cutoff frequency of Embodiment 13.

Embodiment 14 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 13, and thecoefficients used in Embodiment 14 are the same as those shown in Table7, except for the coefficient A (in Embodiment 14, A=1.7). Lens data arethe same as those used in Embodiment 8. The shape of the opticallow-pass filter of Embodiment 14 is such that the shape of the opticallow-pass filter of Embodiment 13 is stretched in the direction of thez-axis.

FIGS. 108, 109, 110 and 111 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 112, 113, 114 and 115 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system. Ascan be seen from FIGS. 111 and 115, if the optical low-pass filter isformed into the aforesaid shape, the cutoff frequency is shifted to thelower frequency than the cutoff frequency of Embodiment 13.

Accordingly, it is possible to readily cope with a modification of thespecifications (the number of pixels) of an image pickup element such asa CCD.

(Embodiment 15)

In Embodiment 15 of the present invention, a shape for providing alow-pass effect is formed at a surface different from the surface r14.

Embodiment 15 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 13, and thecoefficients used in Embodiment 15 are the same as those shown in Table7, except for the coefficient A (in Embodiment 15, A=1.35). The shapeS(ρ, φ) of the optical low-pass filter is added to the aspheric surfacer13 located in the vicinity of the stop 2. Lens data are the same asthose used in Embodiment 8. The shape of the optical low-pass filter ofEmbodiment 15 is such that the shape of the optical low-pass filter ofEmbodiment 13 is stretched in the direction of the z-axis.

FIGS. 116, 117, 118 and 119 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 120, 121, 122 and 123 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system.

As is apparent from the above description, the optical low-pass filtermay be added to any surface that is located in the vicinity of the stop2. As shown in FIG. 74, the flat plate 5 which does not greatly affectthe photographing optical system and to which the optical low-passfilter is added may be provided in the vicinity of the stop 2.

(Embodiment 16)

Embodiment 16 of the present invention is shown in FIG. 124. InEmbodiment 16, the shape of an optical low-pass filter according to thepresent invention is added to the shape of a surface r6 whichconstitutes part of a single-focus lens in the photographing opticalsystem shown in FIG. 124. The photographing optical system includes thestop 2 and the infrared cut-filter 3. Reference numeral 4 denotes a CCD.

Lens data for the photographing optical system are shown in Table 9.

TABLE 9 Radius of Refractive Abbe Surface Curvature Separation IndexNumber 1 2.56858 0.50000 1.620 60.3 2 −14.32671 0.45521 1 3 −2.677430.20000 1.575 41.5 4 1.95389 0.12908 1 5 0.00000 0.10000 1 6 28.281680.20000 1.569 56.3 7 1.26838 0.70000 1.620 60.3 8 −2.23688 1 f 6 Fnumber 3.5 2ω 41.1°

Embodiment 16 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 13, and thecoefficients used in Embodiment 16 are the same as those shown in Table8, except for the coefficient A (in Embodiment 16, A=0.74).

FIGS. 125, 126, 127 and 128 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained in the photographingoptical system including the optical low-pass filter. FIGS. 129, 130,131 and 132 respectively show a wavefront aberration (λ=587.56 nm), arelative point spread (white), a relative line spread (white) and an MTFcurve (white) which are obtained in a photographing optical system(Table 8) which is not provided with the optical low-pass filter. As isapparent from the above description, the optical low- pass filter of thepresent invention is capable of readily coping with different kinds oflens systems having different characteristics (aberrations).

(Embodiment 17)

In Embodiment 17 of the present invention, the setting of the angle δ ofthe shape for providing a low-pass effect is changed.

Embodiment 17 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 12, and thecoefficients used in Embodiment 17 are the same as those shown in Table8, except for the coefficient A (in Embodiment 10, A=0.74). Lens dataare the same as those used in Embodiment 16.

FIGS. 133, 134, 135 and 136 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained in a photographingoptical system including an optical low-pass filter of δ=22.5°.

FIGS. 137, 138, 139 and 140 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained in a photographingoptical system including an optical low-pass filter of δ=0° and A=0.79.

(Embodiment 18)

The shape S(ρ, φ) of the optical low-pass filter of Embodiment 18 of thepresent invention is expressed by the following expressions:$\begin{matrix}{{{S\left( {\rho,\varphi} \right)} = {{R(\rho)} \times \cos \quad \left\{ {2\left( {\varphi + {k\quad \rho^{2}} + \delta} \right)} \right\}}},} & (55) \\{{R(\rho)} = {A\left\{ {{15c\quad \rho^{6}} + {\left( {{{- 20}c} + {4b}} \right)\rho^{4}} + {\left( {a + {6c} - {3b}} \right)\rho^{2}}} \right\} \lambda}} & (56) \\{\quad {= {{A\left( {{a\quad \rho^{2}} + {b\left( {{4\rho^{4}} - {3\rho^{2}}} \right)} + {c\left( {{15\rho^{6}} - {20\rho^{2}} + {6\rho^{2}}} \right)}} \right)}{\lambda.}}}} & (57)\end{matrix}$

The coefficients used in Embodiment 18 are shown in Table 10.

TABLE 10 A = 1. a = 1.451742609  b = −.5237310811 c = .3263970323 δ =1/4π κ = −1/8π

The contour lines of the optical low-pass filter of Embodiment 18(Expression (55)) are shown in FIG. 141, and the variation in shape ofthe optical low-pass filter relative to the rotational direction (the φdirection) for ρ=0.5, 1 is shown in FIG. 142.

FIGS. 143, 144, 145 and 146 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained from a photographingoptical system in which the optical low-pass filter having the aforesaidshape is added to the surface r4 represented by the corresponding lensdata shown in Table 9 of Embodiment 16. As shown in FIG. 146, Embodiment18 can achieve an effect similar to that of the optical low-pass filterof Embodiment 16.

(Embodiment 19)

In Embodiment 19, an optical low-pass filter having a gradientrefractive index is added to the photographing optical system (zoomlens) shown in FIG. 93, similarly to Embodiment 12.

The gradient refractive index N(ρ, φ) of the optical low-pass filter ofEmbodiment 19 is expressed by the following expressions: $\begin{matrix}\begin{matrix}{{N\left( {\rho,\varphi} \right)} = {{N0} + {\delta \quad {N\left( {\rho,\varphi} \right)}}}} \\{{= {{N0} + {{{Nr}(\rho)} \times \cos \quad \left\{ {2\left( {\varphi + {k\quad \rho} + \delta} \right)} \right\}}}},}\end{matrix} & (58)\end{matrix}$

 Nr(ρ)=A(aρ+bρ ² +cp ³)λ.  (59)

The coefficients used in Embodiment 12 are shown in Table 11.

TABLE 11 A =1. a = 6.053535057 δ = 1/4π   b = −9.618170038 c =5.66276797 κ = −1/8π

FIGS. 147, 148, 149 and 150 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 151, 152, 153 and 154 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system.

(Embodiment 20)

The optical low-pass filter of Embodiment 20 is added to the surface r14of the third lens unit shown in FIG. 47 similarly to the opticallow-pass filter of Embodiment 8, but Embodiment 20 differs in shape fromany of the above-described embodiments.

The shape S(ρ, φ) of the optical low-pass filter of Embodiment 20 isexpressed by the following expressions: $\begin{matrix}{{{S\left( {\rho,\varphi} \right)} = {\sum\limits_{m}\quad {{Am} \times {{Rm}(\rho)} \times \cos \quad \left\{ {m\left( {\varphi + {k\quad m\quad \rho} + {\delta \quad m}} \right)} \right\}}}},{m = 2},6,10,} & (60)\end{matrix}$

 R 2(ρ)=(a2ρ+b2ρ² +c2ρ³)λ,  (61)

R 6(ρ)=(a6ρ+b6ρ²)λ,  (62)

R 10(ρ)=(a10ρ+b10ρ²)λ,  (63)

where m is an integer and δm and km are constants.

The coefficients used in Embodiment 20 are shown in Table 12.

TABLE 12 A₂ = A₆ = A₁₀ = 1 K₂ = K₆ = K₁₀ = −π/8 δ₂ = δ₆ = δ₁₀ = π/4 a₂ =−4.538 a₆ = −0.606 a₁₀ = 0.121 b₂ = 9.613  b₆ = −0.238 b₁₀ = 0.048 c₂ =−5.380

The contour lines of the optical low-pass filter of Embodiment 20(Expression (60)) are shown in FIG. 155, and the variation in shape ofthe optical low-pass filter relative to the rotational direction (the φdirection) for ρ=0.5, 1 is shown in FIG. 156.

In the case of the optical low-pass filter which is realized as theamount of variation in shape of the lens, its wavefront aberration is:

W(ρ, φ)=S(ρ, φ)×(1−n),  (64)

where n is the refractive index of the lens.

The wavefront aberration W(ρ, φ) which occurs in the optical low-passfilter of Embodiment 20 is expressed as: $\begin{matrix}{{{W\left( {\rho,\varphi} \right)} = {\sum\limits_{m}\quad {{Am} \times {{Wrm}(\rho)} \times \cos \quad \left\{ {m\left( {\varphi + {{fm}(\rho)} + {\delta \quad m}} \right)} \right\}}}},} & (65)\end{matrix}$

where fm(ρ) is an arbitrary function of ρ, m is an integer not less than2, Am and δm are constants.

FIGS. 157, 158, 159 and 160 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) relative to the x-direction, all ofwhich are obtained on a shorter focal length side of the photographingoptical system including the optical low-pass filter. FIGS. 161, 162,163 and 164 respectively show a wavefront aberration (λ=587.56 nm), arelative point spread (white), a relative line spread (white) and an MTFcurve (white) which are obtained on a longer focal length side of suchphotographing optical system.

As is apparent from the above description, the optical low-pass filterof Embodiment 20 produces a wavefront aberration analogous to its shapeto separate a point image into a plurality of point images in an imageplane so that the value of MTF can be effectively reduced over the rangeof spatial frequencies higher than a predetermined spatial frequency atwhich the value of MTF is made zero. The predetermined spatial frequency(cutoff frequency) at which the value of MTF is made zero is obtainedfrom the pitch of the pixels of an image pickup element such as a CCD tobe used. In Embodiment 13, the pixel pitch is 5 μm, and the cutofffrequency is 100 lines/mm.

(Embodiment 21)

In Embodiment 21 of the present invention, the cutoff frequency is setto a lower frequency than the cutoff frequency of Embodiment 20.

Embodiment 21 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 20, and thecoefficients used in Embodiment 21 are the same as those shown in Table12, except for the coefficient Am (in Embodiment 20, A₁₀=A₆=A₂=1.6).Lens data are the same as those used in Embodiment 8. The shape of theoptical low-pass filter of Embodiment 21 is such that the shape of theoptical low-pass filter of Embodiment 20 is stretched in the directionof the z-axis.

FIGS. 165, 166, 167 and 168 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 169, 170, 171 and 172 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system. Ascan be seen from FIGS. 168 and 172, if the optical low-pass filter isformed into the aforesaid shape, the cutoff frequency is shifted to thelower frequency than the cutoff frequency of Embodiment 13. Accordingly,it is possible to readily cope with a modification of the specifications(the number of pixels) of an image pickup element such as a CCD.

(Embodiment 22)

In Embodiment 22 of the present invention, a shape for providing alow-pass effect is formed at a surface different from the surface r14.

Embodiment 22 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 20, and thecoefficients used in Embodiment 22 are the same as those shown in Table12, except for the coefficient Am (in Embodiment 22, A₁₀=A₆=A₂=0.96).The shape S(ρ, φ) of the optical low-pass filter is added to theaspheric surface r13 disposed in the vicinity of the stop 2. Lens dataare the same as those used in Embodiment 8. The shape of the opticallow-pass filter of Embodiment 22 is such that the shape of the opticallow-pass filter of Embodiment 20 is squeezed by a small amount in thedirection of the z-axis.

FIGS. 173, 174, 175 and 176 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 177, 178, 179 and 180 respectively show awavefront aberration (λ=587.56. nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system. Asis apparent from the above description, the optical low-pass filter maybe added to any surface that is located in the vicinity of the stop 2.As shown in FIG. 74, the flat plate 5 which does not greatly affect thephotographing optical system and to which the optical low-pass filter isadded may be provided in the vicinity of the stop 2.

(Embodiment 23)

In Embodiment 23 of the present invention, the setting of the angle δ ofthe shape for providing a low-pass effect is changed.

Embodiment 23 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 23, and thecoefficients used in Embodiment 23 are the same as those shown in Table12, except for the coefficient Am (in Embodiment 20, A₁₀=A₆=A₂=1.05).Lens data are the same as those used in Embodiment 8.

FIGS. 181, 182, 183 and 184 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of a photographing optical system including an opticallow-pass filter of δ=22.5°. FIGS. 185, 186, 187 and 188 respectivelyshow a wavefront aberration (λ=587.56 nm), a relative point spread(white), a relative line spread (white) and an MTF curve (white) whichare obtained on a longer focal length side of such photographing opticalsystem.

(Embodiment 24)

The shape S(ρ, φ) of the optical low-pass filter of Embodiment 24 of thepresent invention is expressed by the following expressions:

S(ρ, φ)=ΣAm×Rm(ρ)×cos {m(φ+kmρ ² +δm)}, m=2,6,10,  (66)

R 2(ρ)=(b2ρ² +c ²ρ³ +d2ρ⁴ +f2ρ⁶)λ,  (67)

R 6(ρ)=(b6ρ² +c6ρ³ +d6ρ⁴)λ,  (68)

R 10(ρ)=(b10ρ² +c10ρ³ +d10ρ⁴)λ.  (69)

The coefficients used in Embodiment 18 are shown in Table 13.

TABLE 13 A₂ = A₆ = A₁₀ = 1 K₂ = K₆ = K₁₀ = −π/8 δ₂ = δ₆ = δ₁₀ = π/4 b₂ =−3.123 b₆ = −3.069 b₁₀ = 0.614 c₂ = −7.417 c₆ = 3.708  c₁₀ = −0.742 d₂ =19.100 d₆ = −1.535 d₁₀ = 0.307 f₂ = −9.102

The contour lines of the optical low-pass filter of Embodiment 24(Expression (66)) are shown in FIG. 189, and the variation in shape ofthe optical low-pass filter relative to the rotational direction (the φdirection) for ρ=0.5, 1 is shown in FIG. 190.

FIGS. 191, 192, 193 and 194 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of a photographing optical system in which the opticallow-pass filter having the aforesaid shape is added to the surface r4represented by the corresponding lens data shown in Embodiment 8. FIGS.195, 196, 197 and 198 respectively show a wavefront aberration (λ=587.56nm), a relative point spread (white), a relative line spread (white) andan MTF curve (white) which are obtained on a longer focal length side ofsuch photographing optical system.

(Embodiment 25)

In Embodiment 25, an optical low-pass filter having a gradientrefractive index is added to the photographing optical system (zoomlens) shown in FIG. 93, similarly to each of Embodiments 12 and 19.

The gradient refractive index N(ρ, φ) of the optical low-pass filter ofEmbodiment 25 is expressed by the following expressions: $\begin{matrix}{\begin{matrix}{{N\left( {\rho,\varphi} \right)} = {{N0} + {\delta \quad {N\left( {\rho,\varphi} \right)}}}} \\{{= {{N0} + {\sum\quad {{Am} \times {{Nm}(\rho)} \times \cos \quad \left\{ {{m\quad \varphi} + {k\quad m\quad \rho} + {\delta \quad m}} \right\}}}}},}\end{matrix}{{m = 2},6,10,}} & (70)\end{matrix}$

 Nr 2(ρ)=(a2ρ+b2ρ² +c2ρ³)λ,  (71)

Nr 6(ρ)=(a6ρ+b6ρ²)λ,  (72)

Nr 10(ρ)=(b10ρ+b10ρ²)λ.  (73)

The coefficients used in Embodiment 25 are shown in Table 14.

TABLE 14 A₂ = A₆ = A₁₀ = 1 K₂ = K₆ = K₁₀ = −π/8 δ₂ = δ₆ = δ₁₀ = π/4 b₂ =−3.123 b₆ = −3.069 b₁₀ = 0.614 a₂ = −0.528 a₆ = −0.667 a₁₀ = 0.133 b₂ =11.031 b₆ = −0.261 b₁₀ = 0.052 c₂ = −6.186

FIGS. 199, 200, 201 and 202 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 203, 204, 205 and 206 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system.

(Embodiment 26)

Embodiment 26 of the present invention will be described below.

The optical low-pass filter of Embodiment 26 is added to the surface r14of the third lens unit shown in FIG. 47 similarly to the opticallow-pass filter of Embodiment 8, but Embodiment 26 differs in shape fromEmbodiment 8.

The shape S(ρ, φ) of the optical low-pass filter of Embodiment 26 isexpressed by the following expressions:

S(ρ, φ)=R(ρ)×T(φ),  (74)

R(ρ)=1.8λρ,  (75)

T(φ)=cos {2(φ+π/4)}.  (76)

As expressed by Expressions (74) to (76), the shape of the opticallow-pass filter of Embodiment 26 is a shape which is asymmetrical aboutthe axis of the third lens unit and is added to the axisymmetrical shapethereof.

If the respective x- and y-axes of the coordinate system shown in FIG. 3represent the directions of arrangement of the pixels of an image pickupelement such as a CCD to be used with the photographing optical system,a radial direction (cos{2(φ+π/4)}=1) in which a phase advancing orretarding action is large inclines by 45° with respect to each of thedirections of arrangement of the pixels. The contour lines of theoptical low-pass filter (Expression (74)) taken in such radial directionare shown in FIG. 207.

In the case of the optical low-pass filter which is realized as theamount of variation in shape of the lens, its wavefront aberration is:

W(ρ, φ)=S(ρ, φ)×(1−n),  (77)

where n is the refractive index of the lens.

FIGS. 208, 209, 210 and 211 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 212, 213, 214 and 215 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system.

As is apparent from the above description, the optical low-pass filterof Embodiment 26 produces a wavefront aberration analogous to its shapeto separate a point image into a plurality of point images in an imageplane so that the value of MTF can be effectively reduced over the rangeof spatial frequencies higher than a predetermined spatial frequency atwhich the value of MTF is made zero. The predetermined spatial frequency(cutoff frequency) at which the value of MTF is made zero is obtainedfrom the pitch of the pixels of an image pickup element such as a CCD tobe used. In Embodiment 26, the pixel pitch is 5 μm, and the cutofffrequency is 100 lines/mm.

FIG. 216 shows the contour lines of the optical low-pass filter taken ina radial direction in which (cos {4(φ+π/8)}=1) in which its phaseadvancing or retarding action is large if the period of the periodfunction T(φ) relative to the rotational direction is made shorter andif

R(ρ)=1.07λρ,  (78)

T(φ)=cos {4(φ+π/8)}.  (79)

FIGS. 217, 218, 219 and 220 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. As is apparent from the above description, by reducingthe period of the period function T(φ), it is possible to reduce thevalue of MTF over the range of far higher spatial frequencies.

Incidentally, if the wavefront aberration which occurs in a pencil ofrays passing through the optical low-pass filter is expressed by thefollowing expressions:

W(ρ, φ)=Wr(φ)×Wt(φ),  (80)

where Wr(ρ) is a wavefront aberration relative to the radial directionand Wt(φ) is a wavefront aberration relative to the rotationaldirection, the wavefront aberration at a predetermined radial positionof the pencil of rays which has passed through the optical low-passfilter of Embodiment 26 satisfies the following conditions:

 Wr(0.3)/Wr(0.6)>0,  (81)

Wr(0.6)/Wr(0.9)>0.  (82)

In Embodiment 26, since the optical low-pass filter is arranged so thatsuch a wavefront aberration occurs, it is possible to achieve theaforesaid effect.

(Embodiment 27)

Embodiment 27 of the present invention provides an optical low-passfilter having a shape which is set to produce a wavefront aberrationwhich satisfies the following conditions:

Wr′(0.3)/Wr′(0.6)>1,  (83)

Wr′(0.6)/Wr′(0.9)<1.  (84)

The shape S(ρ, φ) of the optical low-pass filter of Embodiment 27 isexpressed by the following expressions:

S(ρ, φ)=R(ρ)×T(φ),  (85)

R(ρ)=A₁(a ₁ ρ+b ₁ρ² +c ₁ρ³)λ,  (86)

T(φ)=cos {2(φ+δ)}.  (87)

The coefficients used in Embodiment 27 are shown in Table 15.

TABLE 15 a₁ = 3.8111 b₁ = −4.7586 c₁ = 2.6334 A₁ = 1 δ = π/4

The contour lines and the cross section of the optical low-pass filter(Expression (86)) taken in a direction in which its phase advancing orretarding action is large are shown in FIGS. 221 and 222, respectively.Lens data are the same as those used in Embodiment 8.

Since the amount of variation in the wavefront aberration with respectto the direction in which the phase advancing or retarding action islarge is proportional to the amount of variation in the shape of theoptical low-pass filter, the amount of variation in the wavefrontaberration is expressed as follows by using the amount of variation inthe shape of the optical low-pass filter with respect to the directionin which the phase advancing or retarding action is large:

Wr(ρ)R(ρ),  (88)

Wr′(ρ)=∂Wr(ρ)/∂ρ∂R(ρ)/∂ρ.  (89)

Therefore, the following conditions are satisfied:

Wr′(0.3)/Wr′(0.6)=1.764>1 ,  (90)

Wr′(0.6)/Wr′(0.9)=0.574<1.  (91)

FIGS. 223, 224, 225 and 226 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained at F1.65 (fullyopen) on a shorter focal length side of the photographing optical systemincluding the optical low-pass filter. FIGS. 227, 228, 229 and 230respectively show a wavefront aberration (λ=587.56 nm), a relative pointspread (white), a relative line spread (white) and an MTF curve (white)which are obtained in the photographing optical system which is set toF2.8. FIGS. 231, 232, 233 and 234 respectively show a wavefrontaberration (λ=587.56 nm), a relative point spread (white), a relativeline spread (white) and an MTF curve (white) which are obtained in thephotographing optical system which is set to F5.6. By satisfying theaforesaid Expressions (83) and (84) in this manner, it is possible toobtain a stable low-pass effect for each F number.

(Embodiment 28)

In Embodiment 28 of the present invention, the cutoff frequency is setto a lower frequency than the cutoff frequency of Embodiment 27.

Embodiment 28 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 27, and thecoefficients used in Embodiment 28 are the same as those shown in Table15, except for the coefficient A (in Embodiment 28, A=1.875). Lens dataare the same as those used in Embodiment 8. The shape of the opticallow-pass filter of Embodiment 28 is such that the shape of the opticallow-pass filter of Embodiment 27 is stretched in the direction of thez-axis (in the direction of the optical axis).

FIGS. 235, 236, 237 and 238 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 239, 240, 241 and 242 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system. Ascan be seen from FIGS. 238 and 242, if the optical low-pass filter isformed into the aforesaid shape, the cutoff frequency is shifted to thelower frequency than the cutoff frequency of Embodiment 27. Accordingly,it is possible to readily cope with a modification of the specifications(the number of pixels) of an image pickup element such as a CCD.

(Embodiment 29)

In Embodiment 29 of the present invention, a shape for providing alow-pass effect is formed at a surface different from the surface r14.

Embodiment 29 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 27, and thecoefficients used in Embodiment 29 are the same as those shown in Table15, except for the coefficient A (in Embodiment 29, A=0.963). The shapeS(ρ, φ) of the optical low-pass filter is added to the aspheric surfacer13 located in the vicinity of the stop 2. Lens data are the same asthose used in Embodiment 8. The shape of the optical low-pass filter ofEmbodiment 29 is such that the shape of the optical low-pass filter ofEmbodiment 26 is squeezed by a small amount in the direction of thez-axis.

FIGS. 243, 244, 245 and 246 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 247, 248, 249 and 250 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system. Asis apparent from the above description, the optical low-pass filter maybe added to any surface that is located in the vicinity of the stop 2.As shown in FIG. 74, the optical low-pass filter may be added to theflat plate 5 which is provided in the vicinity of the stop 2.

(Embodiment 30)

Embodiment 30 of the present invention will be described below. InEmbodiment 30, similarly to Embodiment 16, the shape of an opticallow-pass filter is added to the shape of the surface r6 whichconstitutes part of a single-focus lens in the photographing opticalsystem shown in FIG. 124, but Embodiment 30 differs in shape fromEmbodiment 16.

Embodiment 30 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 27, and thecoefficients used in Embodiment 30 are the same as those used inEmbodiment 16, except for the coefficient A (in Embodiment 30, A=0.75).

FIGS. 251, 252, 253 and 254 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained from thephotographing optical system including the optical low-pass filter. Asis apparent from the above description, the optical low-pass filter ofthe present invention can readily be applied to different kinds of lenssystems having different characteristics (aberrations), by modifying thedesign (or shape) of the optical low-pass filter.

(Embodiment 31)

In Embodiment 31 of the present invention,, the setting of the angle δis changed.

Embodiment 31 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 27, and thecoefficients used in Embodiment 31 are the same as those shown in Table3, except for the coefficient A (in Embodiment 31, A=0.65). Lens dataare the same as those used in Embodiment 16. In Embodiment 31, sinceδ=22.5°, a radial direction (cos{2(φ+δ)}=1) in which its phase advancingor retarding action is large inclines by 22.5° with respect to each ofthe directions of arrangement of the pixels.

FIGS. 255, 256, 257 and 258 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained from thephotographing optical system including the optical low-pass filter.

FIGS. 259, 260, 261 and 262 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained from a photographingoptical system including an optical low-pass filter of A=0.625 and δ=0°.

As is apparent from the above description, by varying the phase δ andthe coefficient A relative to the rotational direction, the cutofffrequency is set to a predetermined spatial frequency so that an effectequivalent to that of Embodiment 29 can be achieved.

(Embodiment 32)

The shape S(ρ, φ) of the optical low-pass filter of Embodiment 32 of thepresent invention is expressed by the following expressions:

S(ρ, φ)=R(ρ)×T(φ),  (92)

$\begin{matrix}{{R(\rho)} = \quad {A_{2}\left( {{a_{2}\rho^{2}} + {b_{2}\left( {{4\rho^{4}} - {3\rho^{2}}} \right)} + {c_{2}\left( {{15\rho^{6}} - {20\rho^{4}} + {6\rho^{2}}} \right)} +} \right.}} & {\quad (93)} \\{{\quad \left. {d_{2}\left( {{56\rho^{8}} + {105\rho^{6}} + {60\rho^{4}} - {10\rho^{2}}} \right)} \right)}\lambda} & \quad \\{= \quad \left\{ {{56A_{2}d_{2}\rho^{8}} + {\left( {{{- 105}A_{2}d_{2}} + {15A_{2}c_{2}}} \right)\rho^{6}} + \left( {{{- 20}A_{2}c_{2}} +} \right.} \right.} & {\quad (94)} \\{{{\quad \left. {{60A_{2}d_{2}} + {4A_{2}b_{2}}} \right)}\rho^{4}} + \left( {{6A_{2}c_{2}} - {10A_{2}d_{2}} + {A_{2}a_{2}} -} \right.} & \quad \\{{\left. {{\quad \left. {3A_{2}b_{2}} \right)}\rho^{6}} \right\} \lambda},} & \quad\end{matrix}$

 T(φ)=cos {2(φ+δ)}.  (95)

The coefficients used in Embodiment 32 are shown in Table 16.

TABLE 16 a₂ = 1.48247238 b₂ = −.495094038 c₂ = .309258846 d₂ =−.190947222 A₂ = 1. δ = π/4

The contour lines and the cross section of the optical low-pass filter(Expression (93)) taken in a direction (cos (2φ+δ)=1) in which its phaseadvancing or retarding action is large are shown in FIGS. 263 and 264,respectively.

Thus, the following conditions are satisfied:

Wr′(0.3)/Wr′(0.6)=9.065>1,  (96)

Wr′(0.6)/Wr′(0.9)=0.172<1.  (97)

FIGS. 265, 266, 267 and 268 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained from a photographingoptical system in which the optical low-pass filter having the aforesaidshape is added to the surface r4 represented by the corresponding lensdata shown in Table 9 of Embodiment 16. As shown in FIG. 268, Embodiment32 can achieve an effect similar to that of the optical low-pass filterof Embodiment 30.

(Embodiment 33)

In Embodiment 33, an optical low-pass filter having a gradientrefractive index is added to the photographing optical system (zoomlens) shown in FIG. 93, similarly to Embodiment 12.

The gradient refractive index N(ρ, φ) of the optical low-pass filter ofEmbodiment 33 is expressed by the following expressions: $\begin{matrix}\begin{matrix}{{N\left( {\rho,\varphi} \right)} = {{N0} + {\delta \quad {N\left( {\rho,\varphi} \right)}}}} \\{{= {{N0} + {{{Nr}(\rho)} \times {{Nt}(\varphi)}}}},}\end{matrix} & (98)\end{matrix}$

 Nr(ρ)=A ₂(a ₂ ρ+b ₂ρ² +c ₂ρ³)λ,  (99)

Nt(φ)=cos (2+δ).  (100)

The coefficients used in Embodiment 33 are shown in Table 17.

TABLE 17 a₂ = 3.8111 b₂ = −4.7586 c₂ = 2.6334 A₂ = 1.125 δ = π/4

In the case of the optical low-pass filter which is realized by giving avariation in refractive index to an optical member, its wavefrontaberration is

W(ρ, φ)=δN(ρ, φ)×d,  (101)

where d is the thickness of the optical low-pass filter.

A wavefront aberration relative to the direction in which the phaseadvancing or retarding action is large is proportional to a functionNr(ρ), and the amount of variation in the wavefront aberration isproportional to the amount of variation in the shape of the opticallow-pass filter. Therefore,

Wr(ρ)Nr(ρ),  (102)

Wr′(ρ)=∂Wr(ρ)/∂ρ∂Nr(ρ)/∂ρ,  (103)

so that the following conditions are satisfied:

Wr′(0.3)/Wr′(0.6)=1.764>1,  (104)

Wr′(0.6)/Wr(0.9)=0.574<1.  (105)

FIGS. 269, 270, 271 and 272 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 273, 274, 275 and 276 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system.

As is apparent from the above description of Embodiment 33, with anoptical member having a gradient refractive index, it is possible togive a wavefront aberration to an incident pencil of rays while theincident pencil of rays is passing through different positions of theoptical member. Accordingly, the optical member can be made to functionas the optical low-pass filter according to the present invention.

(Embodiment 34)

Embodiment 34 of the present invention will be described below. Theoptical low-pass filter of Embodiment 34 is added to the surface r14 ofthe third lens unit shown in FIG. 47 similarly to the optical low-passfilter of Embodiment 8, but Embodiment 34 differs in shape fromEmbodiment 8.

Furthermore, in Embodiment 34, the shape of the optical low-pass filteris set to produce a wavefront aberration which satisfies the followingconditions:

Wr(0.25)/Wr(0.75)<0,  (158)

Wr′(0.3)/Wr′(0.6)<0.  (159)

The shape S(ρ, φ) of the optical low-pass filter of Embodiment 34 isexpressed by the following expressions:

S(ρ, φ)=R(ρ)×T(φ),  (106)

R(ρ)=A ₂(a ₂ ρ+b ₂ρ² +c ₂ρ³ +d ₂ρ⁴ +e ₂ρ⁵)λ,  (107)

T(φ)=cos {2(φ+δ)}.  (108)

The coefficients used in Embodiment 34 are shown in Table 18.

TABLE 18 a₂ = 3.7868 b₂ = 3.0715 c₂ = −14.235 d₂ = −7.5854 e₂ = 13.647A₂ = 1 δ = π/4

The contour lines and the cross section of the optical low-pass filter(Expression (107)) taken in a radial direction in which its phaseadvancing or retarding action is large are shown in FIGS. 277 and 278,respectively.

The optical low-pass filter of Embodiment 34 satisfies the followingconditions:

Wr(0.25)/Wr(0.75)=−1.502<0,  (109)

Wr′(0.3)/Wr′(0.6)=−0.278<0.  (110)

FIGS. 279, 280, 281 and 282 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 283, 289, 285 and 286 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system.

As is apparent from the above description, the optical low-pass filterof Embodiment 34 produces a wavefront aberration analogous to its shapeto separate a point image into a plurality of point images in an imageplane so that the value of MTF can be effectively reduced over the rangeof spatial frequencies higher than a predetermined spatial frequency atwhich the value of MTF is made zero. The predetermined spatial frequency(cutoff frequency) at which the value of MTF is made zero is obtainedfrom the pitch of the pixels of an image pickup element such as a CCD tobe used. In Embodiment 34, the pixel pitch is 5 μm, and the cutofffrequency is 100 lines/mm.

(Embodiment 35)

In Embodiment 35, the cutoff frequency is set to a lower frequency thanthe cutoff frequency of Embodiment 34.

Embodiment 35 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 34, and thecoefficients used in Embodiment 34 are the same as those shown in Table18, except for the coefficient A (in Embodiment 35, A=1.422). Lens dataare the same as those used in Embodiment 8. The shape of the opticallow-pass filter of Embodiment 35 is such that the shape of the opticallow-pass filter of Embodiment 34 is stretched in the direction of thez-axis.

FIGS. 287, 288, 289 and 290 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 291, 292, 293 and 294 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system. Ascan be seen from FIGS. 290 and 294, if the optical low-pass filter isformed into the aforesaid shape, the cutoff frequency is shifted to thelower frequency than the cutoff frequency of Embodiment 34. Accordingly,it is possible to readily cope with a modification of the specifications(the number of pixels) of an image pickup element such as a CCD.

(Embodiment 36)

In Embodiment 36, a shape for providing a low-pass effect is formed at asurface different from the surface r14.

Embodiment 36 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 34, and thecoefficients used in Embodiment 36 are the same as those shown in Table18, except for the coefficient A (in Embodiment 36, A=0.978). The shapeS(ρ, φ) of the optical low-pass filter is added to the aspheric surfacer13 located in the vicinity of the stop 2. Lens data are the same asthose used in Embodiment 8. The shape of the optical low-pass filter ofEmbodiment 36 is such that the shape of the optical low-pass filter ofEmbodiment 34 is squeezed by a small amount in the direction of thez-axis.

FIGS. 295, 296, 297 and 298 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 299, 300, 301 and 302 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system. Asis apparent from the above description, the optical low-pass filter maybe added to any surface that is located in the vicinity of the stop 2.As shown in FIG. 74, the optical low-pass filter may be added to theflat plate 5 which is provided in the vicinity of the stop 2.

(Embodiment 37)

In Embodiment 37, the shape of an optical low-pass filter is added tothe shape of the surface r6 which constitutes part of a single-focuslens in the photographing optical system of Embodiment 16 shown in FIG.124.

Embodiment 37 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 34, and thecoefficients used in Embodiment 37 are the same as those shown in Table18, except for the coefficient A (in Embodiment 37, A=0.462).

FIGS. 303, 304, 305 and 306 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained from thephotographing optical system including the optical low-pass filter. Asis apparent from the above description, the optical low-pass filter ofthe present invention can readily be applied to different kinds of lenssystems having different characteristics (aberrations).

(Embodiment 38)

In Embodiment 38, the setting of the angle δ is changed.

Embodiment 38 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of Embodiment 34, and thecoefficients used in Embodiment 38 are the same as those shown in Table18, except for the coefficient A (in Embodiment 38, A=0.489). Lens dataare the same as those used in Embodiment 16. In Embodiment 38, sinceδ=30°, a radial direction (cos (φ+δ)=1) in which its phase advancing orretarding action is large inclines by 30° with respect to each of thedirections of arrangement of the pixels.

FIGS. 307, 308, 309 and 310 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained from thephotographing optical system including the optical low-pass filter.

FIGS. 311, 312, 313 and 314 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained from a photographingoptical system including an optical low-pass filter of δ=60°.

As is apparent from the above description, by varying the phase δ andthe coefficient A relative to the rotational direction, the cutofffrequency is set to a predetermined spatial frequency so that an effectequivalent to that of Embodiment 37 can be achieved.

(Embodiment 39)

The shape S(ρ, φ) of the optical low-pass filter of Embodiment 39 isexpressed by the following expressions:

S(ρ, φ)=R(ρ)×T(φ),  (111)

$\begin{matrix}{{R(\rho)} = \quad \left\{ {A\left( {{a\quad \rho^{2}} + {{b\left( {{4\quad \rho^{4}} - 3} \right)}\rho^{2}} + {{c\left( {{15\rho^{4}} - {20\quad \rho^{2}} + 6} \right)}\rho^{2}} +} \right.}\quad \right.} & {\quad (112)} \\{{\quad \left. {{d\left( {{56\quad \rho^{6}} + {105\quad \rho^{4}} + {60\quad \rho^{2}} - 10} \right)}\quad \rho^{2}} \right\}}\lambda} & \quad \\{= \quad \left\{ {{56\quad A_{2}d_{2}\rho^{8}} + {\left( {{15\quad A_{2}c_{2}} - {105\quad A_{2}d_{2}}} \right)\rho^{6}} +} \right.} & {\quad (113)} \\{\quad {{\left( {{{- 20}\quad A_{2}c_{2}} + {60\quad A_{2}d_{2}} + {4\quad A_{2}b_{2}}} \right)\rho^{4}} +}} & \quad \\{{{\quad \left. {\left( {{A_{2}a_{2}} + {6\quad A_{2}c_{2}} - {3A_{2}b_{2}} - {10\quad A_{2}d_{2}}} \right)\rho^{2}} \right\}}\quad \lambda},} & \quad\end{matrix}$

 T(φ)=cos {2(φ+δ)}.  (114)

The coefficients used in Embodiment 39 are shown in Table 19.

TABLE 19 a₂ = 0.1342 b₂ = −0.1909 c₂ = 0.1467 d₂ = −0.04997 A₂ = 1 δ =π/4

The contour lines and the cross section of the optical low-pass filter(Expression (112)) taken in a direction (cos{2(φ+δ)}=1) in which thephase advancing or retarding action of Embodiment 39 is large are shownin FIGS. 315 and 316, respectively.

Thus, the following conditions are satisfied:

Wr(0.25)/Wr(0.75)=−1.770<0,  (115)

Wr′(0.3)/Wr′(0.6)=−0.624<0.  (116)

FIGS. 317, 318, 319 and 320 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained from a photographingoptical system in which the optical low-pass filter having the aforesaidshape is added to the surface r4 represented by the corresponding lensdata shown in Table 9 of Embodiment 16. As shown in FIG. 320, Embodiment39 can achieve an effect similar to that of the optical low-pass filterof Embodiment 37.

(Embodiment 40)

In Embodiment 40, an optical low-pass filter having a gradientrefractive index is added to a photographing optical system (zoom lens)in a manner similar to Embodiment 33.

The gradient refractive index N(ρ, φ) of the optical low-pass filter ofEmbodiment 40 is expressed by the following expressions: $\begin{matrix}\begin{matrix}{{N\left( {\rho,\varphi} \right)} = {{N0} + {\delta \quad {N\left( {\rho,\varphi} \right)}}}} \\{{= {{N0} + {{Nr}\quad (\rho) \times {{Nt}(\varphi)}}}},}\end{matrix} & (117)\end{matrix}$

 Nr(ρ)=A ₂(a ₂ ρ+b ₂ρ² +c ₂ρ³ +d ₂ρ⁵)λ,  (118)

Nt(φ)=cos {2(φ+δ)}.  (119)

The coefficients used in Embodiment 40 are shown in Table 20.

TABLE 20 a₂ = 3.7868 b₂ = 3.0715 c₂ = −14.235 d₂ = −7.5854 e₂ = 13.647A₂ = 1.067 δ = π/4

The optical low-pass filter of Embodiment 40 satisfies the followingconditions:

Wr(0.25)/Wr(0.75)=−1.502<0,  (120)

Wr′(03)/Wr′(0.6)=−0.278<0.  (121)

FIGS. 321, 322, 323 and 324 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 325, 326, 327 and 328 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system.

Each of Expressions (81) to (84), (158) and (159) is expressed as theratio of wavefront aberrations relative to the radial direction, butsuch a wavefront aberration is proportional to the radial shape of theoptical low-pass filter if the gradient refractive index thereof isuniform, or to the gradient refractive index if the shape of the opticallow-pass filter is flat.

Therefore, each of Expressions (81) to (84), (158) and (159) may also beexpressed as follows: if the gradient refractive index is uniform,

Wr(0.3)/Wr(0.6)=R(03)/R(0.6)>0,  (81)

Wr(0.6)/Wr(0.9)=R(0.6)/R(0.9)>0,  (82)

Wr′(0.3)/Wr′(0.6)=R′(03)/R′(0.6)>1,  (83)

Wr′(0.6)/Wr′(0.9)=R′(0.6)/R′(0.9)<1,  (84)

Wr(0.25)/Wr(0.75)=R(0.25)/R(0.75)<0,  (158)

Wr′(0.3)/Wr′(0.6)=R′(03)/R′(0.6)<0;  (159)

and if the shape of the optical low-pass filter is flat,

Wr(03)/Wr(0.6)=Nr(0.3)/Nr(0.6)>0,  (81)

Wr(0.6)/Wr(0.9)=Nr(0.6)/Nr(0.9)>0,  (82)

Wr′(0.3)/Wr′(0.6)=Nr′(0.3)/Nr′(0.6)>1,  (83)

 Wr′(0.6)/Wr′(0.9)=Nr′(0.6)/Nr′(0.9)<1,  (84)

Wr(0.25)/Wr(0.75)=Nr(0.25)/Nr(0.75)<0,  (158)

Wr′(0.3)/Wr(0.6)=Nr′(03)/Nr′(0.6)<0,  (159)

where R′(ρ)=∂R(ρ)/∂ρ and Nr′(ρ)=∂Nr(ρ)/∂ρ.

(Embodiment 41)

Embodiment 41 of the present invention will be described below. Theoptical low-pass filter of Embodiment 34 is added to the surface r14 ofthe third lens unit shown in FIG. 47 similarly to the optical low-passfilter of Embodiment 8, but Embodiment 41 differs in shape fromEmbodiment 8.

The shape S(ρ, φ)of the optical low-pass filter of Embodiment 41 isexpressed by the following expressions: $\begin{matrix}{{S\left( {\rho,\varphi} \right)} = \left\{ \begin{matrix}{{{A \cdot {{R2}(\rho)}}\quad \cos \quad {\left\{ {2\left( {\varphi + {\delta \quad 2}} \right)} \right\} \cdot \lambda}},} & {{0 \leq \rho \leq 0.31},} \\{{{A \cdot {{R4}(\rho)}}\quad \cos \quad {\left\{ {4\left( {\varphi + {\delta \quad 4}} \right)} \right\} \cdot \lambda}},} & {0.31{{\leq \rho \leq 1},}}\end{matrix} \right.} & (122)\end{matrix}$

 R 2(ρ)=0.56 sin (πρ′),  (123)

R 4(ρ)=2.969ρ″+2.408ρ″²−11.16ρ″³−5.947ρ″⁴+10.70ρ″⁵,   (124)

A=1, ρ′=ρ/0.31, δ2=π/4, δ4=π/8,

ρ″=(ρ−0.31)/0.69.

The shape of the optical low-pass filter of Embodiment 41 is a shapewhich is asymmetrical about the axis of the third lens unit and is addedto the axisymmetrical shape thereof.

The contour lines of the optical low-pass filter represented byExpression (122) is shown in FIG. 329.

In the case of the optical low-pass filter which is realized as theamount of variation in shape of the lens, its wavefront aberration is

W(ρ, φ)=S(ρ, φ)×(1−n),  (125)

where n is the refractive index of the lens.

Therefore, the wavefront aberration W(ρ, φ) which occurs in thewavefront of the pencil of rays transmitted through the optical low-passfilter of Embodiment 41 is expressed by the following expression:$\begin{matrix}{{W\left( {\rho,\varphi} \right)} = \left\{ \begin{matrix}{{{A \cdot \lambda \cdot \left( {n - 1} \right) \cdot {{R2}(\rho)}}\quad \cos \quad \left\{ {2\left( {\varphi + {\delta \quad 2}} \right)} \right\}},} & {{0 \leq \rho \leq 0.31},} \\{{{A \cdot \lambda \cdot \left( {n - 1} \right) \cdot {{R4}(\rho)}}\quad \cos \quad \left\{ {4\left( {\varphi + {\delta \quad 4}} \right)} \right\}},} & {0.31{{\leq \rho \leq 1},}}\end{matrix} \right.} & (126)\end{matrix}$

In the optical low-pass filter of Embodiment 41, since the period of itsphase advancing and retarding areas is made shorter in its peripherythan in its center, it is possible to achieve a stable low-pass effecteven if a photographing lens system into which to incorporate theoptical low-pass filter has a variable F number.

FIGS. 330, 331, 332 and 333 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained at F1.65 (fullyopen) on a shorter focal length side of the photographing optical systemincluding the optical low-pass filter. FIGS. 334, 335, 336 and 337respectively show a wavefront aberration (λ=587.56 nm), a relative pointspread (white), a relative line spread (white) and an MTF curve (white)which are obtained at F2.8 on the shorter focal length side of thephotographing optical system. FIGS. 338, 339, 340 and 341 respectivelyshow a wavefront aberration (λ=587.56 nm), a relative point spread(white), a relative line spread (white) and an MTF curve (white) whichare obtained at F5.6 on the shorter focal length side of thephotographing optical system. FIGS. 342, 343, 344 and 345 respectivelyshow a wavefront aberration (λ=587.56 nm), a relative point spread(white), a relative line spread (white) and an MTF curve (white) whichare obtained at F1.65 (fully open) on a longer focal length side of thephotographing optical system.

As is apparent from the above description, the optical low-pass filterof Embodiment 41 produces a wavefront aberration analogous to its shapeto separate a point image into a plurality of point images in an imageplane so that the value of MTF can be effectively reduced over the rangeof spatial frequencies higher than a predetermined spatial frequency atwhich the value of MTF is made zero. The predetermined spatial frequency(cutoff frequency) at which the value of MTF is made zero is obtainedfrom the pitch of the pixels of an image pickup element such as a CCD tobe used. In Embodiment 41, the pixel pitch is 5 μm, and the cutofffrequency is 100 lines/mm.

(Embodiment 42)

In Embodiment 42, the cutoff frequency is set to a lower frequency thanthe cutoff frequency of the optical low-pass filter in the photographingoptical system (zoom lens) of Embodiment 41.

The optical low-pass filter of Embodiment 42 uses the same functionalexpressions that express the shape S(ρ, φ) of the optical low-passfilter of Embodiment 41, and the coefficients used in Embodiment 42 arethe same as those used in Expression (122), except for the coefficient A(in Embodiment 42, A=1.4). Lens data are the same as those used inEmbodiment 8. The shape of the optical low-pass filter of Embodiment 42is such that the shape of the optical low-pass filter of Embodiment 41is stretched in the direction of the z-axis (in the direction of theoptical axis).

FIGS. 346, 347, 348 and 349 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 350, 351, 352 and 353 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system. Ascan be seen from FIGS. 349 and 353, if the optical low-pass filter isformed into the shape used in Embodiment 42, the cutoff frequency isshifted to the lower frequency than the cutoff frequency of the opticallow-pass filter of Embodiment 41. Accordingly, it is possible to readilycope with a modification of the specifications (the number of pixels) ofan image pickup element such as a CCD.

(Embodiment 43)

In Embodiment 43, a shape for providing a low-pass effect is formed at asurface different from the surface r14 used in each of Embodiments 41and 42.

Embodiment 43 uses the same functional expressions that express theshape S(ρ, φ) of the optical low-pass filter of the optical low-passfilter of Embodiment 41, but the shape S(ρ, φ) of the optical low-passfilter is added to the aspheric surface r13 located in the vicinity ofthe stop 2. The other lens data are the same as those used in Embodiment8.

FIGS. 354, 355, 356 and 357 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 358, 359, 360 and 361 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system.

As is apparent from the above description, a surface to which to add theoptical low-pass filter is not limited to a specific surface, but it isdesirable to provide the optical low-pass filter in the vicinity of thestop 2. For example, as shown in FIG. 74, the flat plate 5 which doesnot greatly affect the photographing optical system and to which theoptical low-pass filter is added may be provided in the vicinity of thestop 2.

(Embodiment 44)

The shape S(ρ, φ) of the optical low-pass filter of Embodiment 44 of thepresent invention is expressed by the following expressions:

S(ρ, φ)={S 1(ρ, φ)+S 2(ρ, φ)}λ,  (127)

$\begin{matrix}{{{S1}\left( {\rho,\varphi} \right)} = \left\{ {{{\begin{matrix}{{{{R1}(\rho)}\quad \cos \quad \left\{ {2\left( {\varphi + {\delta \quad 2}} \right)} \right\}},} & {{0 \leq \rho \leq 0.335},} \\0 & {{0.335 \leq \rho \leq 1},}\end{matrix}\quad \delta \quad 2} = {\pi/4}},} \right.} & (128)\end{matrix}$

 R1(ρ)=61.64ρ²−197.75ρ³−1113.7ρ⁴+5523.4ρ⁵−8012.9ρ⁶+9798.5ρ⁷−13079.0ρ⁸,  (129)

$\begin{matrix}{{{S2}\left( {\rho,\varphi} \right)} = \left\{ {{{\begin{matrix}{0,} & {{0 \leq \rho < 0.24},} \\{{{R2}\quad (\rho)\quad \cos \quad \left\{ {4\left( {\varphi + {\delta \quad 4}} \right)} \right\}},} & {{0.24 \leq \rho \leq 1},}\end{matrix}\quad \delta \quad 4} = {\pi/8}},} \right.} & (130)\end{matrix}$

 R2(ρ)=79.562ρ′²−603.3ρ′³+1704.8ρ′⁴−1679.3ρ′⁵−1105.9ρ′⁶+3187.3ρ′⁷−1607.6ρ′⁸,

ρ′=ρ−0.24.  (131)

The contour lines of the optical low-pass filter (Expression (127)) areshown in FIGS. 362.

FIGS. 363, 364, 365 and 366 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained at F1.65 (fullyopen) on a shorter focal length side of a photographing optical system,in which system the optical low-pass filter having the aforesaid shapeis added to the surface r14 represented by the corresponding lens datashown in Embodiment 8. FIGS. 367, 368, 369 and 370 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedat F2.8 on the shorter focal length side of the photographing opticalsystem. FIGS. 371, 372, 373 and 374 respectively show a wavefrontaberration (λ=587.56 nm), a relative point spread (white), a relativeline spread (white) and an MTF curve (white) which are obtained at F5.6on the shorter focal length side of the photographing optical system.FIGS. 375, 376, 377 and 378 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained at F1.65 (fullyopen) on a longer focal length side of the photographing optical system.As is apparent from the above description, the optical low-pass filterof Embodiment 44 can achieve an effect similar to that of the opticallow-pass filter of Embodiment 41.

(Embodiment 45)

In Embodiment 45, an optical low-pass filter having a gradientrefractive index is added to the photographing optical system (zoomlens) shown in FIG. 93, similarly to Embodiment 12.

The gradient refractive index N(ρ, φ) of the optical low-pass filter ofEmbodiment 45 is expressed by the following expressions:

N(ρ, φ)=N 0+δN(ρ, φ),  (132)

$\begin{matrix}{{\delta \quad {N\left( {\rho,\varphi} \right)}} = \left\{ {{{\begin{matrix}{{{{Nr2}(\rho)}\quad \cos \quad \left( {{2\varphi} + {\delta \quad 2}} \right)\lambda},} & {{0 \leq \rho \leq 0.31},} \\{{{{Nr4}(\rho)}\quad \cos \quad \left( {{4\varphi} + {\delta \quad 4}} \right)\lambda},} & {{0.31 \leq \rho \leq 1},}\end{matrix}\quad \delta \quad 2} = {\pi/4}},{{\delta 4} = {\pi/8}},} \right.} & (133)\end{matrix}$

 Nr 2(ρ)=0.64 sin (πρ′),ρ′=ρ/0.31,  (134)

Nr 4(ρ)=3.414ρ″+2.769ρ″²−12.83ρ″³−6.839ρ″⁴+12.305ρ″⁵,

ρ″=(ρ−0.31)/0.69.  (135)

In the case of the optical low-pass filter which is realized by giving avariation in refractive index to an optical member, its wavefrontaberration is

W(ρ, φ)=δN(ρ, φ)×d,  (136)

where d is the thickness of the optical low-pass filter.

FIGS. 379, 380, 381 and 382 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of the photographing optical system including the opticallow-pass filter. FIGS. 383, 384, 385 and 386 respectively show awavefront aberration (λ=587.56 nm), a relative point spread (white), arelative line spread (white) and an MTF curve (white) which are obtainedon a longer focal length side of such photographing optical system.

As is apparent from the above description of Embodiment 45, with anoptical member having a gradient refractive index, it is possible togive a wavefront aberration to an incident pencil of rays while theincident pencil of rays is passing through different positions of theoptical member. Accordingly, the optical member can be made to functionas the optical low-pass filter according to the present invention.

In each of the optical low-pass filters of Embodiments 41 to 45 of thepresent invention, since the period of the area having a phase advancingaction and that of the area having a phase retarding actions are variedwith respect to the radial direction (the p direction), it is possibleto effectively decrease MTF relative to high-frequency components evenif the stop 2 is placed in any state from a fully open state to thestate of a maximum reduced aperture, and it is also possible to realizea low-pass effect on various kinds of stops having different apertureshapes.

(Embodiment 46)

In the optical low-pass filter of Embodiment 46 is, its basic shape tobe represented by a continuous function is approximated by a step-formedshape having a step size smaller than the wavelength of light. The shapeof the optical low-pass filter of Embodiment 46 is expressed by thefollowing expression using a step function which converts a continuousfunction into a step-formed shape:

S(ρ, φ)=Step(A×R(ρ)×T(φ)).  (137)

The step function is, for example, a function which converts acontinuous function into a step-formed shape having a ¼ pitch of thewavelength of light.

The shape of the optical low-pass filter of Embodiment 46 is added tothe flat plate 1 provided in the vicinity of the stop 2 in thephotographing optical system shown in FIG. 6. The contour lines whichrepresent the shape of the optical low-pass filter of Embodiment 46 areshown in FIG. 387. In FIG. 387, the region defined by each contour linecorresponds to a step size of ¼ wavelength, and the portion between eachcontour line and an adjacent contour line is flat. In the shape of theoptical low-pass filter shown in FIG. 387, with respect to the center ofthe opening portion, each portion marked “+” is projected to a maximumdegree, whereas each portion marked “−” is dented to a maximum degree.

As shown in FIG. 387, if the shape of the optical low-pass filter ofEmbodiment 46 of the present invention is represented by a cylindricalcoordinate system the origin of which corresponds to the center of theopening portion, the shape of the optical low-pass filter stepwisechanges in the rotational direction from a portion having a phaseadvancing action on the phase of an incident wavefront (any of theportions marked “−”) to a portion having a phase retarding action on thephase of the incident wavefront (the adjacent one of the portions marked“+”).

A numerical example of the basic shape of the surface of the opticallow-pass filter according to Embodiment 46 is shown below:

S 1(ρ, φ)=A 1×R 1(ρ)×cos 2(φ),  (138)

where A1 is a constant and λ is a wavelength.

R 1(ρ)=(3.534ρ+2.867ρ²−13.267ρ³−7.079ρ⁴+12.737ρ⁵)λ, 0≦ρ≦1, 0≦φ2π.  (139)

In the optical low-pass filter of Embodiment 46, portions which havephase advancing actions on the phase of an incident wavefront andportions which have phase retarding actions on the phase of the incidentwavefront are provided in the radial direction as well. The radial basicshape for φ=0 is shown in FIG. 388, and the actual cross-sectional shapeof the optical low-pass filter is shown in FIG. 389.

FIG. 390 shows the contour lines of the point spread in an image plane,and FIG. 391 shows the line spread obtained at F1.65 by performing anaddition in a direction perpendicular to the pixel-array direction of aCCD. FIGS. 392 and 393 show the respective MTF curves of the opticallow-pass filter of Embodiment 46 which is set to F1.65 and F5.6.

The continuous shape of the optical low-pass filter of any otherembodiment can be approximated by a step-formed shape such as that ofthe optical low-pass filter of Embodiment 46, and the opticalperformance obtained from the step-formed shape is approximatelyequivalent to that obtained from the continuous shape.

(Embodiment 47)

FIG. 394 shows Embodiment 47 of the present invention. The opticalsystem of Embodiment 47 is composed of a single molded lens L10 made ofa PMMA material, and focuses a single wavelength onto the CCD 4. Lensdata are shown in Table 21.

TABLE 21 Radius of Refractive Abbe Surface Curvature Separation IndexNumber  1* 22.85257 5.00000 1.49171 57.4 2 −130.97850 f = 40   F2.0

The shape of the reference surface of a surface r1 and the rotationallysymmetrical aspheric terms of the surface r1 are expressed by thefollowing expressions: $\begin{matrix}{{{{S0}(\rho)} = \frac{\left( {r\quad \rho} \right)^{2}/R}{1 + \sqrt{1 - {\left( {1 + k} \right)\left( {r\quad {\rho/R}} \right)^{2}}}}},} & (140)\end{matrix}$

 ASP(ρ)=a(rρ)⁴ +b(rρ)⁶ +c(rρ)⁸ +d(rρ)¹⁰,  (141)

where R is a radius of the osculating surface and k, a, b, c and d areaspheric coefficients.

The aspheric coefficients are shown in Table 22.

TABLE 22 k a b c d −7.83171e − 01 −2.09422e − 07 −2.66250e − 09 0 0

In Embodiment 47, a shape which has an optical low-pass action (alow-pass shape) is added to the surface r1 so that the lens L10 itselfcan be made to function as an optical low-pass filter.

This low-pass shape (V(ρ, φ)) is expressed by the following expressions:

V(ρ, φ)=R(ρ)cos {2(φ+π4)},  (142)

 R(ρ)=A _(v)(a _(v) ρ+b _(v)ρ² +c _(v)ρ³ +d _(v)ρ⁴ =e _(v)ρ⁵)λ.  (143)

Each coefficient of the low-pass shape of Embodiment 47 is shown inTable 23.

TABLE 23 A_(υ) = 1 a_(υ) = 3.16, b_(υ) = 2.58, c_(υ) = −11.9, d_(υ) =−6.34 e_(υ) = 11.4

The contour lines of the low-pass shape expressed by Expressions (142)and (143) are shown in FIG. 395.

The wavefront aberration given to a passing pencil of rays by thelow-pass shape is

W(ρ, φ)=V(ρ, φ)×(1−n),  (144)

where n is the refractive index of the lens L10.

FIGS. 396, 397, 398 and 399 respectively show a wavefront aberration, arelative point spread, a relative line spread and an MTF curve which areobtained from the optical system (λ=587.56 nm) of Embodiment 47. FIGS.400, 401, 402 and 403 respectively show a wavefront aberration, arelative point spread, a relative line spread and an MTF curve which areobtained from an optical system in which no low-pass shape is formed atthe surface r1.

As is apparent from the above description, the optical system ofEmbodiment 47 produces a wavefront aberration analogous to the low-passshape to separate a point image into a plurality of point images in animage plane so that the value of MTF can be effectively reduced over therange of spatial frequencies higher than a predetermined spatialfrequency at which the value of MTF is made zero. The predeterminedspatial frequency (cutoff frequency) at which the value of MTF is madezero is obtained from the pitch of the pixels of an image pickup elementsuch as a CCD to be used. In Embodiment 47, the pixel pitch is 5 μm, andthe cutoff frequency is 100 lines/mm.

However, a lens actually molded on the basis of such design valuesundergoes complicated deformation and non-uniform distribution of aninner refractive index owing to molding conditions and the like. FIGS.404 to 406 respectively show in contour line an example of deformationof the surface r1 (E1), an example of deformation of a surface r2 (E2),and an example of non-uniform distribution of the inner refractive index(E3).

Such error shape can be approximated by using Zernike's polynomialrepresented by $\begin{matrix}\begin{matrix}{{E\left( {\rho,\varphi} \right)} = \quad {{c1} + {{c2}\quad {\rho cos}\quad (\varphi)} + {{c3}\quad \rho \quad \sin \quad (\varphi)} + {{c4}\quad \rho^{2}\cos \quad \left( {2\varphi} \right)} +}} \\{\quad {{{c5}\left( {{2\quad \rho^{2}} - 1} \right)} + {{c6}\quad \rho^{2}\sin \quad \left( {2\varphi} \right)} + {{c7}\quad \rho^{3}\cos \quad \left( {3\varphi} \right)} +}} \\{\quad {{{{c8}\left( {{3\rho^{3}} - {2\rho}} \right)}\cos \quad (\varphi)} + {{{c9}\left( {{3\rho^{3}} - {2\rho}} \right)}\sin \quad (\varphi)} +}} \\{\quad {{{c10}\quad \rho^{3}\sin \quad \left( {3\varphi} \right)} + {{c11}\quad \rho^{4}\cos \quad \left( {4\varphi} \right)} +}} \\{\quad {{{c12}\quad \left( {{4\quad \rho^{4}} - {3\rho^{2}}} \right)\cos \quad \left( {2\varphi} \right)} + {{c13}\left( {{6\rho^{4}} - {6\rho^{2}} + 1} \right)} +}} \\{\quad {{{{c14}\left( {{4\rho^{4}} - {3\rho^{2}}} \right)}\quad \sin \quad \left( {2\varphi} \right)} + {{c15}\quad \rho^{4}\sin \quad \left( {4\varphi} \right)} +}} \\{\quad {{{c16}\quad \rho^{5}\cos \quad \left( {5\quad \varphi} \right)} + {{c17}\quad \left( {{5\rho^{5}} - {4\rho^{3}}} \right)\cos \quad \left( {3\varphi} \right)} +}} \\{\quad {{{c18}\quad \left( {{10\rho^{5}} - {12\quad \rho^{3}} + {3\rho}} \right)\quad \cos \quad (\varphi)} +}} \\{\quad {{{c19}\quad \left( {{10\rho^{5}} - {12\quad \rho^{3}} + {3\rho}} \right)\sin \quad (\varphi)} +}} \\{\quad {{{{c20}\left( {{5\rho^{5}} - {4\rho^{3}}} \right)}\sin \quad \left( {3\varphi} \right)} + {{c21}\quad \rho^{5}\sin \quad \left( {5\varphi} \right)} + \ldots}}\end{matrix} & (145)\end{matrix}$

Tables 24, 25 and 26 respectively show the coefficients obtained byapproximating the shapes shown in FIGS. 404 to 406 by using Expression(145).

TABLE 24 c1 −0.3538 c8  0.0384 c15 −0.0129 c2 0.1128 c9  0.0606 c16−0.0243 c3 0.0657 c10 0.1647 c17 −0.3672 c4 −0.2735 c11 −0.1535 c180.0795 c5 0.0875 c12 0.0331 c19 0.0048 c6 −0.0593 c13 0.0028 c20 −0.0113c7 0.0580 c14 −0.0060 c21 −0.0045

TABLE 25 c1 −0.7858 c8  0.1091 c15 0 c2 0.4593 c9  −0.00329 c16 0 c30.2183 c10 0 c17 0 c4 −0.8197 c11 0 c18 0 c5 −0.3799 c12 0 c19 0 c60.5056 c13 0 c20 0 c7 0.2297 c14 0 c21 0

TABLE 26 c1 −0.0175 c8 0.0024 c15 0.0035 c2 0 c9 0.0028 c16 0 c3 −0.0058c10 0 c17 0 c4 −0.0147 c11 −0.0023 c18 0 c5 0.1097 c12 −0.0029 c19 0 c60 c13 0 c20 0 c7 0 c14 0 c21 0

The respective deformations (E1 and E2) of the surfaces r1 and r2 areobtained by measuring the wavefronts reflected at the surfaces r1 and r2by means of an interferometer, and the non-uniformity (E3) of the innergradient refractive index is obtained by measuring a transmittedwavefront by means of an interferometer while taking E1 and E2 intoaccount.

Such a molding error can be prevented by slowly molding the lens L10 insuch a manner as to prevent non-uniformity from occurring in itsrefractive index, while correcting the shapes of the respective surfacesr1 and r2, but this molding method incurs an increase in cost. For thisreason, in accordance with Embodiment 47 of the present invention, theaforesaid low-pass shape and a shape for correcting the molding errorare simultaneously formed at the surface r1 to which to add theaforesaid low-pass shape.

The shape for correcting the molding error will be described below.Since the shape of the surface r1 is to be corrected, the error (E1) ofthe surface r1 is corrected by adding a shape having signs opposite tothe signs of the shape shown in FIG. 404 to the molding shape of a mold.The error (E2) of the surface r2 is corrected by setting the amount oferror of the surface r2 and redesigning the surface r1 on the basis ofthe amount of error. The non-uniformity of the inner gradient refractiveindex (E3) is corrected by setting the amount of error of a medium orapproximating the inner gradient refractive index by converting theinner gradient refractive index into the amount of error of the shape ofthe surface r1 or r2, and redesigning the surface r1 on the basis of theamount of error which has been set in this manner. The mold is preparedby adding the low-pass shape shown in FIG. 395 to the thus-redesignedshape of the surface r1. The error occurring during molding is fullycorrected by molding an optical low-pass filter by using such mold.

FIG. 407 shows the contour lines of a transmitted wavefront obtainablewhen the amounts of errors E2 and E3 shown in FIGS. 405 and 406 areadded to the surface r2. The shape of the surface r1 which is redesignedwithin the degree of freedom of the surface r1 for the purpose ofcorrecting such a wavefront is described below. In the followingdescription, the aspheric terms other than the low-pass shape arerepresented by Zernike's polynomial Z(ρ, φ) which containsaxisymmetrical terms. Therefore,

ASP(ρ)+H(ρ, φ)→Z(ρ, φ),  (146)

and a shape S1(ρ, φ) of the surface r1 which does not contain thelow-pass shape is expressed as

S 1(ρ, φ)=S 0(ρ)+Z(ρ, φ),  (147)

$\begin{matrix}{{{{S0}(\rho)} = \frac{\rho^{2}/R}{1 + \sqrt{1 - {\left( {1 + k} \right)\left( {\rho/R} \right)^{2}}}}},} & (148) \\\begin{matrix}{{Z\left( {\rho,\varphi} \right)} = \quad {{c1} + {{c2}\quad {\rho cos}\quad (\varphi)} + {{c3}\quad \rho \quad \sin \quad (\varphi)} + {{c4}\quad \rho^{2}\cos \quad \left( {2\varphi} \right)} +}} \\{\quad {{{c5}\left( {{2\quad \rho^{2}} - 1} \right)} + {{c6}\quad \rho^{2}\sin \quad \left( {2\varphi} \right)} + {{c7}\quad \rho^{3}\cos \quad \left( {3\varphi} \right)} +}} \\{\quad {{{{c8}\left( {{3\rho^{3}} - {2\rho}} \right)}\cos \quad (\varphi)} + {{{c9}\left( {{3\rho^{3}} - {2\rho}} \right)}\sin \quad (\varphi)} +}} \\{\quad {{{c10}\quad \rho^{3}\sin \quad \left( {3\varphi} \right)} + {{c11}\quad \rho^{4}\cos \quad \left( {4\varphi} \right)} +}} \\{\quad {{{c12}\quad \left( {{4\quad \rho^{4}} - {3\rho^{2}}} \right)\cos \quad \left( {2\varphi} \right)} + {{c13}\left( {{6\rho^{4}} - {6\rho^{2}} + 1} \right)} +}} \\{\quad {{{{c14}\left( {{4\rho^{4}} - {3\rho^{2}}} \right)}\quad \sin \quad \left( {2\varphi} \right)} + {{c15}\quad \rho^{4}\sin \quad \left( {4\varphi} \right)} +}} \\{\quad {{{c16}\quad \rho^{4}\cos \quad \left( {5\quad \varphi} \right)} + {{c17}\quad \left( {{5\rho^{5}} - {4\rho^{3}}} \right)\cos \quad \left( {3\varphi} \right)} +}} \\{\quad {{{c18}\quad \left( {{10\rho^{5}} - {12\quad \rho^{3}} + {3\rho}} \right)\quad \cos \quad (\varphi)} +}} \\{\quad {{{c19}\quad \left( {{10\rho^{5}} - {12\quad \rho^{3}} + {3\rho}} \right)\sin \quad (\varphi)} +}} \\{\quad {{{{c20}\left( {{5\rho^{5}} - {4\rho^{3}}} \right)}\sin \quad \left( {3\varphi} \right)} + {{c21}\quad \rho^{5}\sin \quad \left( {5\varphi} \right)} + \ldots}}\end{matrix} & (149)\end{matrix}$

The coefficients contained in the above expressions are shown in Table27.

TABLE 27 K = −0.842  c1 0.8452  c2 −0.0551  c3 −0.0436  c4 0.8452  c5−0.4465  c6 −0.0629  c7 −0.0276  c8 −0.0218  c9 0 c10 −0.0013 c11 0.0244c12 0 c13 0 c14 0 c15 0 c16 0 c17 0 c18 0 c19 0 c20 0 c21 0

A deviation from a spheric surface having the shape expressed byExpression (147) is shown in FIG. 408 in contour line.

Therefore, the actual shape S(ρ, φ) of the surface r1 which includes thelow-pass shape is expressed as $\begin{matrix}\begin{matrix}{{S\left( {\rho,\varphi} \right)} = {{{S1}\left( {\rho,\varphi} \right)} + {V\left( {\rho,\varphi} \right)}}} \\{= {{{S0}(\rho)} + {Z\left( {\rho,\varphi} \right)} + {{V\left( {\rho,\varphi} \right)}.}}}\end{matrix} & (150)\end{matrix}$

Contour lines which represent the deviation of the surface r1 from thespheric surface are shown in FIG. 409.

Since the surface r1 of the optical low-pass filter is molded in such amanner that a correction shape having the amount of error E1 iscancelled in the shape of the surface r1, by a mold based on the shapeexpressed by Expression (150), the shape of the surface r1 becomes anaspheric shape expressed by subtracting the correcting shape having theamount of error E1 from the reference surface shape S0(ρ), the low-passshape V(ρ, φ) and Z(ρ, φ). In other words, the surface r1 of the opticallow-pass filter, when it is finished, is formed of only correctionshapes having the amounts of errors E2 and E3.

(Embodiment 48)

In Embodiment 48, a low-pass shape is added to the surface r14 of thephotographing optical system shown in FIG. 47. The low-pass shape formedat the surface r14 is expressed by the following expressions using acylindrical coordinate system; $\begin{matrix}{{{V\left( {\rho,\varphi} \right)} = {\sum\limits_{m}{{{AmRm}(\rho)}\quad \cos \quad \left\{ {2\left( {\varphi - {{\pi/8}\quad \rho} + {\pi/4}} \right)} \right\}}}},{m = 2},6,10,} & (151)\end{matrix}$

 R 2(ρ)=A2(a2ρ+b2ρ² +c2ρ³)λ,  (152)

R 6(ρ)=A6(a6ρ+b6ρ²)λ,  (153)

R 10(ρ)=A10(a10ρ+b10ρ²)λ.  (154)

The coefficents used in Embodiment 48 are shown in Table 28.

TABLE 28 A₂ = A₆ = A₁₀ = 1 a₂ = −4.538 a₆ = −0.606 a₁₀ = 0.121 b₂ =9.613 b₆ = −0.238 b₁₀ = 0.048 c₂ = −5.380

The contour lines of the low-pass shape of Embodiment 48 are shown inFIG. 410.

FIGS. 411, 412, 413 and 414 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained on a shorter focallength side of a zoom lens using the design values of Embodiment 48.FIGS. 415, 416, 417 and 418 respectively show a wavefront aberration(λ=587.56 nm), a relative point spread (white), a relative line spread(white) and an MTF curve (white) which are obtained from a longer focallength side of such zoom lens.

In Embodiment 48, a correction shape for correcting the error of a lenswhich occurs during molding is added to the surface r14 in a mannersimilar to that described above in connection with Embodiment 47. Theshape S(ρ, φ) of the surface r14 which is set in the mold is expressedby the following expressions:

S(ρ, φ)=S 0(ρ)+Z(ρ, φ)+V(ρ, φ),  (155)

$\begin{matrix}{{{{S0}(\rho)} = \frac{\left( {r\quad \rho} \right)^{2}/R}{1 + \sqrt{1 - \left( {r\quad {\rho/R}} \right)^{2}}}},} & (156) \\\begin{matrix}{{Z\left( {\rho,\varphi} \right)} = \quad {{{- {.00397}}\quad \rho \quad \cos \quad (\varphi)} - {{.0927}\quad \rho \quad \sin \quad (\varphi)} +}} \\{\quad {{{.410}\quad \rho^{2}\cos \quad \left( {2\varphi} \right)} + {{.190}\quad \rho^{2}\sin \quad \left( {2\varphi} \right)} - {{.253}\quad \rho^{3}\cos \quad \left( {3\varphi} \right)} +}} \\{\quad {{{.115}\left( {{3\varphi^{3}} - {2\quad \rho}} \right)\quad \sin \quad (\varphi)} - {{.0546}\quad \rho^{3}\sin \quad \left( {3\varphi} \right)} +}} \\{\quad {{.0165}\quad \left( {{4\rho^{4}} - {3\rho^{2}}} \right)\quad \cos \quad {\left( {2\varphi} \right).}}}\end{matrix} & (157)\end{matrix}$

A deviation from the spheric shape of the surface r14 expressed byExpression (155) is shown in FIG. 419 in contour line.

As is apparent from the above description of each of Embodiments 47 and48 of the present invention, since a shape for correcting a shape errorwhich occurs during the molding of an optical low-pass filter is addedto a surface at which a low-pass shape is formed, the optical low-passfilter can exhibit a good low-pass effect and optical performanceequivalent to design values.

(Embodiment 49)

FIG. 420 shows an example in which a photographing optical system (zoomlens) according to any of the above- described embodiments is applied toan optical apparatus. The optical apparatus shown in FIG. 420 includes aphotographing optical system 100 according to any of the above-describedembodiments, the CCD 4 and a recording part 200. A light image comingfrom a subject is formed on the CCD 4 by the photographing opticalsystem 100, and the image signal produced in the CCD 4 is sent to therecording part 200. An observer can observe in a viewfinder opticalsystem (not shown) an image displayed on image display means (notshown).

As is apparent from the above description, it is possible to obtain agood image by means of an inexpensive apparatus and arrangement byemploying any of the optical low-pass filters of the disclosedembodiments of the present invention in an optical apparatus such as avideo camera or a digital camera.

What is claimed is:
 1. A method of manufacturing an optical low-passfilter, comprising the steps of: charging a material into a mold; andremoving the material molded by the mold, said optical low-pass filteralternately including a phase advancing area which advances a phase of awavefront of an incident pencil of rays with respect to a phase of awavefront of a center of the incident pencil of rays, and a phaseretarding area which retards the phase of the wavefront of the incidentpencil of rays with respect to the phase of the wavefront of the centerof the incident pencil of rays, said molding having a shape whichcorrects an error occurring during molding of said optical low-passfilter, and said phase advancing area and said phase retarding areabeing alternately formed in a generally non-linear pattern, and in arotational direction centered at an origin corresponding to the centerof the incident pencil of rays.
 2. A method of manufacturing an opticallow-pass filter according to claim 1, wherein the error includes anerror of a shape of said optical low-pass filter which occurs during themolding thereof.
 3. A method of manufacturing an optical low-pass filteraccording to claim 1, wherein the error includes a non-uniformity of arefractive index of said optical low-pass filter which occurs during themolding thereof.
 4. A method of manufacturing an optical low-pass filteraccording to claim 1, wherein the material includes a synthetic resinmaterial.
 5. A method of manufacturing an optical low-pass filteraccording to claim 1, wherein the material includes a glass material.